L(s) = 1 | + 2·5-s − 9-s − 4·11-s − 8·19-s − 25-s + 4·29-s − 4·31-s − 20·41-s − 2·45-s + 14·49-s − 8·55-s − 20·59-s − 20·61-s + 20·79-s + 81-s − 12·89-s − 16·95-s + 4·99-s + 12·101-s − 20·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1/3·9-s − 1.20·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s − 0.718·31-s − 3.12·41-s − 0.298·45-s + 2·49-s − 1.07·55-s − 2.60·59-s − 2.56·61-s + 2.25·79-s + 1/9·81-s − 1.27·89-s − 1.64·95-s + 0.402·99-s + 1.19·101-s − 1.91·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9212009156\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9212009156\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.462160978066871438026940903876, −8.992247727913780046444213259006, −8.575061287422460351624873477467, −8.331817346623484330724800720054, −7.84516834054008749336122989111, −7.55113677523494949792544764678, −6.96235739660030672274847327578, −6.42419391350038918513559925594, −6.39628168566273034171906592994, −5.66649479667331458701397138910, −5.54401764690280223287852598128, −4.94544230669417925357158104974, −4.59923563234529681227119936514, −4.12626618595381125873674830290, −3.42530497345019846639118815560, −2.98968926830801104369618395748, −2.49675245619892744213448618082, −1.89150095134574137367069808941, −1.61910902725255531069007658603, −0.32449252636191658331878872977,
0.32449252636191658331878872977, 1.61910902725255531069007658603, 1.89150095134574137367069808941, 2.49675245619892744213448618082, 2.98968926830801104369618395748, 3.42530497345019846639118815560, 4.12626618595381125873674830290, 4.59923563234529681227119936514, 4.94544230669417925357158104974, 5.54401764690280223287852598128, 5.66649479667331458701397138910, 6.39628168566273034171906592994, 6.42419391350038918513559925594, 6.96235739660030672274847327578, 7.55113677523494949792544764678, 7.84516834054008749336122989111, 8.331817346623484330724800720054, 8.575061287422460351624873477467, 8.992247727913780046444213259006, 9.462160978066871438026940903876