| L(s) = 1 | + 24.5·2-s − 49.9·3-s + 90.0·4-s − 1.22e3·6-s + 8.70e3·7-s − 1.03e4·8-s − 1.71e4·9-s − 8.29e4·11-s − 4.50e3·12-s + 2.85e4·13-s + 2.13e5·14-s − 3.00e5·16-s − 3.74e5·17-s − 4.21e5·18-s − 3.61e5·19-s − 4.35e5·21-s − 2.03e6·22-s + 2.31e6·23-s + 5.17e5·24-s + 7.00e5·26-s + 1.84e6·27-s + 7.83e5·28-s − 6.49e5·29-s + 4.32e6·31-s − 2.06e6·32-s + 4.14e6·33-s − 9.19e6·34-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.356·3-s + 0.175·4-s − 0.386·6-s + 1.37·7-s − 0.893·8-s − 0.873·9-s − 1.70·11-s − 0.0626·12-s + 0.277·13-s + 1.48·14-s − 1.14·16-s − 1.08·17-s − 0.946·18-s − 0.636·19-s − 0.488·21-s − 1.85·22-s + 1.72·23-s + 0.318·24-s + 0.300·26-s + 0.667·27-s + 0.240·28-s − 0.170·29-s + 0.841·31-s − 0.347·32-s + 0.608·33-s − 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.126269036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126269036\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 24.5T + 512T^{2} \) |
| 3 | \( 1 + 49.9T + 1.96e4T^{2} \) |
| 7 | \( 1 - 8.70e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.29e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.74e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.49e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.32e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.21e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.59e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.08e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.37e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.29e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.54e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.37e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.89e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.53e8T + 7.60e17T^{2} \) |
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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
−10.48595741226226355264665030692, −8.778524717815618208958860432733, −8.339733751696881563529951174913, −6.98916855991784061139726191724, −5.75028776172735047247883440437, −5.07591426645797221744761090433, −4.51330760652882265894542930616, −3.05481687386237152873191995160, −2.18848374653434841095859679537, −0.52176922792296559128903886984,
0.52176922792296559128903886984, 2.18848374653434841095859679537, 3.05481687386237152873191995160, 4.51330760652882265894542930616, 5.07591426645797221744761090433, 5.75028776172735047247883440437, 6.98916855991784061139726191724, 8.339733751696881563529951174913, 8.778524717815618208958860432733, 10.48595741226226355264665030692
