| L(s) = 1 | − 27.3·2-s − 81·3-s + 237.·4-s + 625·5-s + 2.21e3·6-s + 2.40e3·7-s + 7.51e3·8-s + 6.56e3·9-s − 1.71e4·10-s − 9.14e4·11-s − 1.92e4·12-s − 7.33e4·13-s − 6.57e4·14-s − 5.06e4·15-s − 3.27e5·16-s + 4.45e5·17-s − 1.79e5·18-s + 4.98e5·19-s + 1.48e5·20-s − 1.94e5·21-s + 2.50e6·22-s − 1.05e6·23-s − 6.09e5·24-s + 3.90e5·25-s + 2.00e6·26-s − 5.31e5·27-s + 5.69e5·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.463·4-s + 0.447·5-s + 0.698·6-s + 0.377·7-s + 0.649·8-s + 0.333·9-s − 0.541·10-s − 1.88·11-s − 0.267·12-s − 0.712·13-s − 0.457·14-s − 0.258·15-s − 1.24·16-s + 1.29·17-s − 0.403·18-s + 0.878·19-s + 0.207·20-s − 0.218·21-s + 2.27·22-s − 0.783·23-s − 0.374·24-s + 0.200·25-s + 0.861·26-s − 0.192·27-s + 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 27.3T + 512T^{2} \) |
| 11 | \( 1 + 9.14e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.33e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.45e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.05e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.43e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.75e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.43e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.43e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.63e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.64e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.16e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.96e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.61e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.58e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.27e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.07e9T + 7.60e17T^{2} \) |
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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
−11.02965206176554671426563635561, −10.18558794327179800506988676031, −9.557876159042840983838327260600, −7.985441439536605435004717105584, −7.50159938193639292012110761261, −5.72133114470839120995784560999, −4.76920781973455673580245982478, −2.56314904915299764996316143633, −1.14978793354320743683906337796, 0,
1.14978793354320743683906337796, 2.56314904915299764996316143633, 4.76920781973455673580245982478, 5.72133114470839120995784560999, 7.50159938193639292012110761261, 7.985441439536605435004717105584, 9.557876159042840983838327260600, 10.18558794327179800506988676031, 11.02965206176554671426563635561
