| L(s) = 1 | − 9.77·2-s − 73.6·3-s − 32.4·4-s + 542.·5-s + 719.·6-s + 343·7-s + 1.56e3·8-s + 3.23e3·9-s − 5.29e3·10-s − 5.39e3·11-s + 2.39e3·12-s + 2.19e3·13-s − 3.35e3·14-s − 3.99e4·15-s − 1.11e4·16-s + 1.84e4·17-s − 3.16e4·18-s − 4.42e4·19-s − 1.76e4·20-s − 2.52e4·21-s + 5.27e4·22-s + 4.38e4·23-s − 1.15e5·24-s + 2.15e5·25-s − 2.14e4·26-s − 7.74e4·27-s − 1.11e4·28-s + ⋯ |
| L(s) = 1 | − 0.863·2-s − 1.57·3-s − 0.253·4-s + 1.93·5-s + 1.36·6-s + 0.377·7-s + 1.08·8-s + 1.48·9-s − 1.67·10-s − 1.22·11-s + 0.399·12-s + 0.277·13-s − 0.326·14-s − 3.05·15-s − 0.681·16-s + 0.908·17-s − 1.27·18-s − 1.48·19-s − 0.492·20-s − 0.595·21-s + 1.05·22-s + 0.750·23-s − 1.70·24-s + 2.76·25-s − 0.239·26-s − 0.757·27-s − 0.0959·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8245213885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8245213885\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 + 9.77T + 128T^{2} \) |
| 3 | \( 1 + 73.6T + 2.18e3T^{2} \) |
| 5 | \( 1 - 542.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.39e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.42e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.98e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.29e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.50e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.55e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.50e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.30e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.58e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.33e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.73e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.93e6T + 8.07e13T^{2} \) |
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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
−12.88331915110836682219309260194, −11.11166414410223734205282508360, −10.35811280442932356797199148826, −9.777590740314024995194578313053, −8.390266298286211418454182749355, −6.74605590007369034408582659169, −5.57513618811131016258180207116, −4.95643115112012180194369415639, −1.93617478239699795212098804694, −0.73285204090141573717617691639,
0.73285204090141573717617691639, 1.93617478239699795212098804694, 4.95643115112012180194369415639, 5.57513618811131016258180207116, 6.74605590007369034408582659169, 8.390266298286211418454182749355, 9.777590740314024995194578313053, 10.35811280442932356797199148826, 11.11166414410223734205282508360, 12.88331915110836682219309260194
