| L(s) = 1 | − 6·4-s − 27·9-s + 66·11-s − 13·16-s + 116·19-s + 440·29-s + 496·31-s + 162·36-s + 312·41-s − 396·44-s + 750·49-s − 1.09e3·59-s + 828·61-s + 644·64-s − 1.82e3·71-s − 696·76-s + 1.08e3·79-s + 486·81-s − 1.58e3·89-s − 1.78e3·99-s − 4.55e3·101-s − 6.45e3·109-s − 2.64e3·116-s + 2.54e3·121-s − 2.97e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/4·4-s − 9-s + 1.80·11-s − 0.203·16-s + 1.40·19-s + 2.81·29-s + 2.87·31-s + 3/4·36-s + 1.18·41-s − 1.35·44-s + 2.18·49-s − 2.41·59-s + 1.73·61-s + 1.25·64-s − 3.04·71-s − 1.05·76-s + 1.54·79-s + 2/3·81-s − 1.88·89-s − 1.80·99-s − 4.48·101-s − 5.66·109-s − 2.11·116-s + 1.90·121-s − 2.15·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.35890703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.35890703\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 - p T )^{6} \) |
| good | 2 | \( 1 + 3 p T^{2} + 49 T^{4} - 17 p^{4} T^{6} + 49 p^{6} T^{8} + 3 p^{13} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 750 T^{2} + 57705 p T^{4} - 169444420 T^{6} + 57705 p^{7} T^{8} - 750 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 7610 T^{2} + 28729095 T^{4} - 72595578540 T^{6} + 28729095 p^{6} T^{8} - 7610 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 24246 T^{2} + 265282351 T^{4} - 1673289769268 T^{6} + 265282351 p^{6} T^{8} - 24246 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 58 T + 9081 T^{2} - 861164 T^{3} + 9081 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 37130 T^{2} + 625555295 T^{4} - 7809452400140 T^{6} + 625555295 p^{6} T^{8} - 37130 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 220 T + 58859 T^{2} - 10101400 T^{3} + 58859 p^{3} T^{4} - 220 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 - 8 p T + 50781 T^{2} - 5187088 T^{3} + 50781 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 48770 T^{2} + 5577662727 T^{4} - 228480272860860 T^{6} + 5577662727 p^{6} T^{8} - 48770 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 156 T + 100167 T^{2} - 24516984 T^{3} + 100167 p^{3} T^{4} - 156 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 372278 T^{2} + 63874841543 T^{4} - 6453481309521044 T^{6} + 63874841543 p^{6} T^{8} - 372278 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 533306 T^{2} + 126813100463 T^{4} - 17054137241744108 T^{6} + 126813100463 p^{6} T^{8} - 533306 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 369602 T^{2} + 67874889383 T^{4} - 10412208947567036 T^{6} + 67874889383 p^{6} T^{8} - 369602 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 548 T + 11419 p T^{2} + 226302104 T^{3} + 11419 p^{4} T^{4} + 548 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 - 414 T - 389 T^{2} + 154404524 T^{3} - 389 p^{3} T^{4} - 414 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 1207890 T^{2} + 688739462775 T^{4} - 250114025195213020 T^{6} + 688739462775 p^{6} T^{8} - 1207890 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 912 T + 1171621 T^{2} + 649961952 T^{3} + 1171621 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 1751090 T^{2} + 1405618993167 T^{4} - 680662783430797020 T^{6} + 1405618993167 p^{6} T^{8} - 1751090 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 542 T + 1317893 T^{2} - 445950836 T^{3} + 1317893 p^{3} T^{4} - 542 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 1248042 T^{2} + 1103016399495 T^{4} - 630549898371298636 T^{6} + 1103016399495 p^{6} T^{8} - 1248042 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 790 T - 3625 T^{2} - 827778380 T^{3} - 3625 p^{3} T^{4} + 790 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 3108986 T^{2} + 5195517820623 T^{4} - 5815064414148144108 T^{6} + 5195517820623 p^{6} T^{8} - 3108986 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.14683414185642512479679067533, −4.72900577533175402130504185335, −4.63987811616725456123901048353, −4.37429123430493752225928330783, −4.26486558659487439820287666440, −4.20015006078409455046818128372, −4.18431131880339789659203921616, −3.94242892453941661332561963177, −3.59320612105945100773166326472, −3.53427035484322211936704981873, −3.19113403497388354539600519829, −2.83073795781045484529770970591, −2.78316261502009388736625661354, −2.74450066485110859435988442785, −2.68744476850135246790732910097, −2.65367924798567292715516736935, −1.87469471772198786620980330762, −1.77105345417228666139525434763, −1.61863219666894656830595526358, −1.35308911784514062930252593028, −1.03882447543936260588392575268, −0.931091076678150464713588190043, −0.51543036041311502224827911407, −0.49894755797449731922978876321, −0.47687039029893101136449582455,
0.47687039029893101136449582455, 0.49894755797449731922978876321, 0.51543036041311502224827911407, 0.931091076678150464713588190043, 1.03882447543936260588392575268, 1.35308911784514062930252593028, 1.61863219666894656830595526358, 1.77105345417228666139525434763, 1.87469471772198786620980330762, 2.65367924798567292715516736935, 2.68744476850135246790732910097, 2.74450066485110859435988442785, 2.78316261502009388736625661354, 2.83073795781045484529770970591, 3.19113403497388354539600519829, 3.53427035484322211936704981873, 3.59320612105945100773166326472, 3.94242892453941661332561963177, 4.18431131880339789659203921616, 4.20015006078409455046818128372, 4.26486558659487439820287666440, 4.37429123430493752225928330783, 4.63987811616725456123901048353, 4.72900577533175402130504185335, 5.14683414185642512479679067533
Plot not available for L-functions of degree greater than 10.
