| L(s) = 1 | + 33.6·2-s + 231.·3-s + 623.·4-s + 224.·5-s + 7.80e3·6-s + 2.40e3·7-s + 3.75e3·8-s + 3.39e4·9-s + 7.55e3·10-s + 4.01e4·11-s + 1.44e5·12-s − 2.85e4·13-s + 8.09e4·14-s + 5.19e4·15-s − 1.92e5·16-s + 3.05e5·17-s + 1.14e6·18-s − 9.12e4·19-s + 1.39e5·20-s + 5.56e5·21-s + 1.35e6·22-s + 6.13e5·23-s + 8.70e5·24-s − 1.90e6·25-s − 9.62e5·26-s + 3.31e6·27-s + 1.49e6·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.65·3-s + 1.21·4-s + 0.160·5-s + 2.45·6-s + 0.377·7-s + 0.324·8-s + 1.72·9-s + 0.238·10-s + 0.827·11-s + 2.01·12-s − 0.277·13-s + 0.562·14-s + 0.264·15-s − 0.734·16-s + 0.888·17-s + 2.57·18-s − 0.160·19-s + 0.195·20-s + 0.624·21-s + 1.23·22-s + 0.457·23-s + 0.535·24-s − 0.974·25-s − 0.413·26-s + 1.20·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.169022634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.169022634\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 33.6T + 512T^{2} \) |
| 3 | \( 1 - 231.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 224.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.01e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.05e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.12e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.13e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.14e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.37e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.38e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.96e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.54e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.18e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.54e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.85e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.28e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.60e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.85e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.16e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.48e9T + 7.60e17T^{2} \) |
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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
−12.60840014927154332122087825043, −11.66659754957314894782245594165, −9.944607722414130244445013511931, −8.859922555190769804761577634179, −7.71728726855131107989309481823, −6.40369601868652500037503654147, −4.84608330273263500114314435150, −3.75435096215460025204479868041, −2.86790849540851471855587587987, −1.65713004771621425858433958945,
1.65713004771621425858433958945, 2.86790849540851471855587587987, 3.75435096215460025204479868041, 4.84608330273263500114314435150, 6.40369601868652500037503654147, 7.71728726855131107989309481823, 8.859922555190769804761577634179, 9.944607722414130244445013511931, 11.66659754957314894782245594165, 12.60840014927154332122087825043
