| L(s) = 1 | − 2.56·2-s − 177.·3-s − 505.·4-s − 625·5-s + 454.·6-s + 2.60e3·8-s + 1.18e4·9-s + 1.60e3·10-s + 6.14e4·11-s + 8.97e4·12-s + 1.24e5·13-s + 1.10e5·15-s + 2.52e5·16-s + 3.78e5·17-s − 3.03e4·18-s − 7.23e5·19-s + 3.15e5·20-s − 1.57e5·22-s + 5.50e5·23-s − 4.62e5·24-s + 3.90e5·25-s − 3.17e5·26-s + 1.39e6·27-s − 3.62e6·29-s − 2.84e5·30-s − 9.99e6·31-s − 1.98e6·32-s + ⋯ |
| L(s) = 1 | − 0.113·2-s − 1.26·3-s − 0.987·4-s − 0.447·5-s + 0.143·6-s + 0.224·8-s + 0.601·9-s + 0.0506·10-s + 1.26·11-s + 1.24·12-s + 1.20·13-s + 0.565·15-s + 0.961·16-s + 1.10·17-s − 0.0681·18-s − 1.27·19-s + 0.441·20-s − 0.143·22-s + 0.410·23-s − 0.284·24-s + 0.200·25-s − 0.136·26-s + 0.504·27-s − 0.952·29-s − 0.0640·30-s − 1.94·31-s − 0.333·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8410530927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8410530927\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.56T + 512T^{2} \) |
| 3 | \( 1 + 177.T + 1.96e4T^{2} \) |
| 11 | \( 1 - 6.14e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.24e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.78e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.50e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.85e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.53e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.52e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.08e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.77e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.08e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.34e9T + 7.60e17T^{2} \) |
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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
−10.80206041857004819452617616579, −9.434262635526617204999795394434, −8.714290386933416191665124305034, −7.51879056468725155373922639714, −6.22056948025931053820358057285, −5.56370759611844179226084202684, −4.31438927912138700026873868355, −3.63236121623922028486813591962, −1.35336730623670078019319495982, −0.52896795033934380443309579938,
0.52896795033934380443309579938, 1.35336730623670078019319495982, 3.63236121623922028486813591962, 4.31438927912138700026873868355, 5.56370759611844179226084202684, 6.22056948025931053820358057285, 7.51879056468725155373922639714, 8.714290386933416191665124305034, 9.434262635526617204999795394434, 10.80206041857004819452617616579
