| L(s) = 1 | + 8·2-s + 78.4·3-s + 64·4-s − 293.·5-s + 627.·6-s − 1.42e3·7-s + 512·8-s + 3.96e3·9-s − 2.34e3·10-s + 5.02e3·12-s + 1.29e4·13-s − 1.13e4·14-s − 2.30e4·15-s + 4.09e3·16-s + 1.19e4·17-s + 3.17e4·18-s + 4.89e4·19-s − 1.87e4·20-s − 1.11e5·21-s + 3.23e4·23-s + 4.01e4·24-s + 7.83e3·25-s + 1.03e5·26-s + 1.39e5·27-s − 9.11e4·28-s + 1.94e4·29-s − 1.84e5·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s − 1.04·5-s + 1.18·6-s − 1.57·7-s + 0.353·8-s + 1.81·9-s − 0.741·10-s + 0.838·12-s + 1.63·13-s − 1.11·14-s − 1.75·15-s + 0.250·16-s + 0.590·17-s + 1.28·18-s + 1.63·19-s − 0.524·20-s − 2.63·21-s + 0.554·23-s + 0.593·24-s + 0.100·25-s + 1.15·26-s + 1.36·27-s − 0.785·28-s + 0.148·29-s − 1.24·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.210902881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.210902881\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 78.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 293.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.42e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 1.29e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.19e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.89e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.94e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.80e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.53e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.90e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.18e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.61e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.65e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91370210280566472603207233313, −9.640944226322607796403924178375, −8.925269151503432241786738792873, −7.78955057753866446906724457212, −7.12818856446513415744564463682, −5.81718214192150077724117070173, −3.89632273687448566916143139133, −3.54397521465226715949172829153, −2.76040018906893522995926330065, −1.01749785592757976185392142189,
1.01749785592757976185392142189, 2.76040018906893522995926330065, 3.54397521465226715949172829153, 3.89632273687448566916143139133, 5.81718214192150077724117070173, 7.12818856446513415744564463682, 7.78955057753866446906724457212, 8.925269151503432241786738792873, 9.640944226322607796403924178375, 10.91370210280566472603207233313
