| L(s) = 1 | + 34.1·2-s + 79.6·3-s + 655.·4-s + 2.72e3·6-s − 2.40e3·7-s + 4.88e3·8-s − 1.33e4·9-s + 6.93e4·11-s + 5.21e4·12-s − 1.05e5·13-s − 8.20e4·14-s − 1.68e5·16-s − 5.68e5·17-s − 4.55e5·18-s − 3.96e5·19-s − 1.91e5·21-s + 2.36e6·22-s + 6.20e5·23-s + 3.89e5·24-s − 3.61e6·26-s − 2.63e6·27-s − 1.57e6·28-s + 4.87e6·29-s − 1.42e6·31-s − 8.25e6·32-s + 5.52e6·33-s − 1.94e7·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.567·3-s + 1.27·4-s + 0.857·6-s − 0.377·7-s + 0.421·8-s − 0.677·9-s + 1.42·11-s + 0.726·12-s − 1.02·13-s − 0.570·14-s − 0.642·16-s − 1.65·17-s − 1.02·18-s − 0.697·19-s − 0.214·21-s + 2.15·22-s + 0.462·23-s + 0.239·24-s − 1.55·26-s − 0.952·27-s − 0.483·28-s + 1.28·29-s − 0.277·31-s − 1.39·32-s + 0.810·33-s − 2.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 34.1T + 512T^{2} \) |
| 3 | \( 1 - 79.6T + 1.96e4T^{2} \) |
| 11 | \( 1 - 6.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.05e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.20e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.31e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.09e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.20e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.02e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.47e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.09e9T + 7.60e17T^{2} \) |
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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
−10.91311945991211688873955907289, −9.344772685129490715066985952676, −8.632978553807167871806733239267, −6.91870475751686756886432430136, −6.29482439980768693608136196393, −4.93920056826668455712181757304, −4.02181651458375474891486032350, −2.99431502674022672929511749371, −2.05349547033056550107787523715, 0,
2.05349547033056550107787523715, 2.99431502674022672929511749371, 4.02181651458375474891486032350, 4.93920056826668455712181757304, 6.29482439980768693608136196393, 6.91870475751686756886432430136, 8.632978553807167871806733239267, 9.344772685129490715066985952676, 10.91311945991211688873955907289
