| L(s) = 1 | + 7·4-s + 10·11-s + 37·16-s + 100·19-s − 230·29-s − 230·31-s + 1.16e3·41-s + 70·44-s + 452·49-s − 760·59-s − 304·61-s − 405·64-s + 80·71-s + 700·76-s − 2.02e3·79-s − 2.04e3·89-s + 6.33e3·101-s − 5.32e3·109-s − 1.61e3·116-s − 2.13e3·121-s − 1.61e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 0.274·11-s + 0.578·16-s + 1.20·19-s − 1.47·29-s − 1.33·31-s + 4.41·41-s + 0.239·44-s + 1.31·49-s − 1.67·59-s − 0.638·61-s − 0.791·64-s + 0.133·71-s + 1.05·76-s − 2.88·79-s − 2.42·89-s + 6.23·101-s − 4.68·109-s − 1.28·116-s − 1.60·121-s − 1.16·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.71267531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.71267531\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7 T^{2} + 3 p^{2} T^{4} + 145 p^{2} T^{6} + 3 p^{8} T^{8} - 7 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 452 T^{2} + 259712 T^{4} - 83658146 T^{6} + 259712 p^{6} T^{8} - 452 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( ( 1 - 5 T + 1105 T^{2} + 17950 T^{3} + 1105 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 13 | \( 1 - 11595 T^{2} + 58956990 T^{4} - 168291031471 T^{6} + 58956990 p^{6} T^{8} - 11595 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 16669 T^{2} + 128845131 T^{4} - 696765837278 T^{6} + 128845131 p^{6} T^{8} - 16669 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 50 T + 206 p T^{2} - 317888 T^{3} + 206 p^{4} T^{4} - 50 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 24177 T^{2} + 422258235 T^{4} - 6413047586710 T^{6} + 422258235 p^{6} T^{8} - 24177 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 115 T + 25759 T^{2} - 830870 T^{3} + 25759 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 + 115 T + 60141 T^{2} + 5913626 T^{3} + 60141 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 20616 T^{2} + 3107453304 T^{4} + 44864752461914 T^{6} + 3107453304 p^{6} T^{8} - 20616 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 580 T + 296575 T^{2} - 83865640 T^{3} + 296575 p^{3} T^{4} - 580 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 126873 T^{2} + 12309399387 T^{4} - 1329823991586454 T^{6} + 12309399387 p^{6} T^{8} - 126873 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 103609 T^{2} + 3919180371 T^{4} - 611394181179878 T^{6} + 3919180371 p^{6} T^{8} - 103609 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 616918 T^{2} + 181410991767 T^{4} - 33107955883012340 T^{6} + 181410991767 p^{6} T^{8} - 616918 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 380 T + 588745 T^{2} + 150882920 T^{3} + 588745 p^{3} T^{4} + 380 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 + 152 T + 593468 T^{2} + 63933158 T^{3} + 593468 p^{3} T^{4} + 152 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 1316576 T^{2} + 829402539632 T^{4} - 314698227099711638 T^{6} + 829402539632 p^{6} T^{8} - 1316576 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 40 T + 396361 T^{2} + 187438400 T^{3} + 396361 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 1902768 T^{2} + 1636568077872 T^{4} - 814870399949877454 T^{6} + 1636568077872 p^{6} T^{8} - 1902768 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 + 1013 T + 1531332 T^{2} + 908300089 T^{3} + 1531332 p^{3} T^{4} + 1013 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2754966 T^{2} + 3495061602231 T^{4} - 2562175992353487892 T^{6} + 3495061602231 p^{6} T^{8} - 2754966 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 1020 T + 2161707 T^{2} + 1313072760 T^{3} + 2161707 p^{3} T^{4} + 1020 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 4888512 T^{2} + 10390398210432 T^{4} - 12341794210097197246 T^{6} + 10390398210432 p^{6} T^{8} - 4888512 p^{12} T^{10} + p^{18} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.39316382378083743076235224012, −5.03268829739542667404139080346, −4.87136103419188486907960287166, −4.49567327556628627583179645057, −4.41347508091482497493025034901, −4.35381101742943076546122528262, −4.07085471921563623218222317302, −3.92813040836608617422081410410, −3.80462684146270974992960793031, −3.71189172593715866074471781757, −3.23937958933816394037819488616, −3.11443642796747224789452551097, −2.83573671523656158284193952664, −2.76722453821194036146583946864, −2.69941418085353006708999976196, −2.62355865962679078980611210546, −1.96526773079395764424878919358, −1.83416561076099519951102046025, −1.67466663559137733344811168422, −1.54704079918297502198978882231, −1.49515164926447225204229737772, −0.75055701109972471154351309615, −0.72051974713804041412184733083, −0.51549971028572369068865599237, −0.37847079169619703003743954056,
0.37847079169619703003743954056, 0.51549971028572369068865599237, 0.72051974713804041412184733083, 0.75055701109972471154351309615, 1.49515164926447225204229737772, 1.54704079918297502198978882231, 1.67466663559137733344811168422, 1.83416561076099519951102046025, 1.96526773079395764424878919358, 2.62355865962679078980611210546, 2.69941418085353006708999976196, 2.76722453821194036146583946864, 2.83573671523656158284193952664, 3.11443642796747224789452551097, 3.23937958933816394037819488616, 3.71189172593715866074471781757, 3.80462684146270974992960793031, 3.92813040836608617422081410410, 4.07085471921563623218222317302, 4.35381101742943076546122528262, 4.41347508091482497493025034901, 4.49567327556628627583179645057, 4.87136103419188486907960287166, 5.03268829739542667404139080346, 5.39316382378083743076235224012
Plot not available for L-functions of degree greater than 10.
