L(s) = 1 | + 18.6·2-s − 27·3-s + 219.·4-s + 184.·5-s − 503.·6-s + 1.63e3·7-s + 1.69e3·8-s + 729·9-s + 3.43e3·10-s + 5.17e3·11-s − 5.91e3·12-s − 974.·13-s + 3.03e4·14-s − 4.97e3·15-s + 3.61e3·16-s + 2.75e4·17-s + 1.35e4·18-s + 2.11e4·19-s + 4.04e4·20-s − 4.40e4·21-s + 9.64e4·22-s + 3.57e4·23-s − 4.58e4·24-s − 4.41e4·25-s − 1.81e4·26-s − 1.96e4·27-s + 3.57e5·28-s + ⋯ |
L(s) = 1 | + 1.64·2-s − 0.577·3-s + 1.71·4-s + 0.659·5-s − 0.950·6-s + 1.79·7-s + 1.17·8-s + 0.333·9-s + 1.08·10-s + 1.17·11-s − 0.988·12-s − 0.123·13-s + 2.95·14-s − 0.380·15-s + 0.220·16-s + 1.36·17-s + 0.549·18-s + 0.708·19-s + 1.12·20-s − 1.03·21-s + 1.93·22-s + 0.612·23-s − 0.677·24-s − 0.565·25-s − 0.202·26-s − 0.192·27-s + 3.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(8.794985266\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.794985266\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
good | 2 | \( 1 - 18.6T + 128T^{2} \) |
| 5 | \( 1 - 184.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.63e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.17e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 974.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.75e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.11e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.90e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.71e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.82e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.12e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.30e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.29e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.36e7T + 8.07e13T^{2} \) |
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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
−10.21372861886667216510135839681, −8.988556155356257268332973667574, −7.68149549375683318234295455421, −6.77800810378015250803252692309, −5.66968122412446962243933474225, −5.23740790925258550977512193115, −4.38970193849905789702191230190, −3.35656039305042897549029002058, −1.88715480193421832589967147175, −1.22438185139693793712090540951,
1.22438185139693793712090540951, 1.88715480193421832589967147175, 3.35656039305042897549029002058, 4.38970193849905789702191230190, 5.23740790925258550977512193115, 5.66968122412446962243933474225, 6.77800810378015250803252692309, 7.68149549375683318234295455421, 8.988556155356257268332973667574, 10.21372861886667216510135839681