| L(s) = 1 | − 17·3-s − 25·5-s + 49·7-s + 208·9-s + 73·11-s − 23·13-s + 425·15-s − 263·17-s − 833·21-s + 625·25-s − 2.15e3·27-s − 1.15e3·29-s − 1.24e3·33-s − 1.22e3·35-s + 391·39-s − 5.20e3·45-s − 3.45e3·47-s + 2.40e3·49-s + 4.47e3·51-s − 1.82e3·55-s + 1.01e4·63-s + 575·65-s + 1.00e4·71-s + 9.50e3·73-s − 1.06e4·75-s + 3.57e3·77-s − 1.21e4·79-s + ⋯ |
| L(s) = 1 | − 1.88·3-s − 5-s + 7-s + 2.56·9-s + 0.603·11-s − 0.136·13-s + 17/9·15-s − 0.910·17-s − 1.88·21-s + 25-s − 2.96·27-s − 1.37·29-s − 1.13·33-s − 35-s + 0.257·39-s − 2.56·45-s − 1.56·47-s + 49-s + 1.71·51-s − 0.603·55-s + 2.56·63-s + 0.136·65-s + 1.99·71-s + 1.78·73-s − 1.88·75-s + 0.603·77-s − 1.94·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6992309621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6992309621\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 17 T + p^{4} T^{2} \) |
| 11 | \( 1 - 73 T + p^{4} T^{2} \) |
| 13 | \( 1 + 23 T + p^{4} T^{2} \) |
| 17 | \( 1 + 263 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 + 1153 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 + 3457 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( 1 - 10078 T + p^{4} T^{2} \) |
| 73 | \( 1 - 9502 T + p^{4} T^{2} \) |
| 79 | \( 1 + 12167 T + p^{4} T^{2} \) |
| 83 | \( 1 + 6382 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 + 3383 T + p^{4} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61056713774949746738759963430, −9.442990711529591803989597583914, −8.204011916923566694736751676407, −7.24172930872485356347968156593, −6.55342212444548457774557813173, −5.42495313467857070278926024555, −4.65330541327905063877911487968, −3.91249897072355486828308187536, −1.68309590380970303927994272146, −0.51422426435272746011843392917,
0.51422426435272746011843392917, 1.68309590380970303927994272146, 3.91249897072355486828308187536, 4.65330541327905063877911487968, 5.42495313467857070278926024555, 6.55342212444548457774557813173, 7.24172930872485356347968156593, 8.204011916923566694736751676407, 9.442990711529591803989597583914, 10.61056713774949746738759963430
