| L(s) = 1 | − 2·4-s + 2·13-s + 4·16-s − 46·19-s + 32·25-s − 94·31-s − 110·37-s + 46·43-s − 4·52-s − 208·61-s − 8·64-s − 194·67-s − 130·73-s + 92·76-s + 226·79-s − 208·97-s − 64·100-s − 238·103-s − 98·109-s + 80·121-s + 188·124-s + 127-s + 131-s + 137-s + 139-s + 220·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/13·13-s + 1/4·16-s − 2.42·19-s + 1.27·25-s − 3.03·31-s − 2.97·37-s + 1.06·43-s − 0.0769·52-s − 3.40·61-s − 1/8·64-s − 2.89·67-s − 1.78·73-s + 1.21·76-s + 2.86·79-s − 2.14·97-s − 0.639·100-s − 2.31·103-s − 0.899·109-s + 0.661·121-s + 1.51·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.48·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07433264945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07433264945\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 80 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4400 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3026 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 104 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 113 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12896 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 104 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.55453481640419041740529738903, −9.505713386376568673210476808105, −9.222054175908024045832977661098, −8.910182480217836198499580298705, −8.607746552631514427123925111892, −8.176084068298713639108636445033, −7.57262719675420339795077380394, −7.04640739208492592152124389790, −6.92654382232964623444551127333, −6.10542686613735438204102033904, −5.89210002556530629649368849919, −5.29473893408680961965871669588, −4.76948962132770955315423845377, −4.35557174064098322875808167754, −3.83706515057428053077967521764, −3.32598395922472054956012873035, −2.72177873511208712586506191645, −1.80454228260531207055942104922, −1.54796115841814924399104948350, −0.087164595751584953522891153827,
0.087164595751584953522891153827, 1.54796115841814924399104948350, 1.80454228260531207055942104922, 2.72177873511208712586506191645, 3.32598395922472054956012873035, 3.83706515057428053077967521764, 4.35557174064098322875808167754, 4.76948962132770955315423845377, 5.29473893408680961965871669588, 5.89210002556530629649368849919, 6.10542686613735438204102033904, 6.92654382232964623444551127333, 7.04640739208492592152124389790, 7.57262719675420339795077380394, 8.176084068298713639108636445033, 8.607746552631514427123925111892, 8.910182480217836198499580298705, 9.222054175908024045832977661098, 9.505713386376568673210476808105, 10.55453481640419041740529738903
