| L(s) = 1 | − 13.1·2-s + 1.72e3·3-s − 8.01e3·4-s − 5.02e4·5-s − 2.27e4·6-s − 2.94e5·7-s + 2.13e5·8-s + 1.39e6·9-s + 6.60e5·10-s − 2.79e5·11-s − 1.38e7·12-s + 3.87e6·14-s − 8.69e7·15-s + 6.28e7·16-s − 1.55e8·17-s − 1.83e7·18-s − 1.94e8·19-s + 4.03e8·20-s − 5.09e8·21-s + 3.67e6·22-s − 1.15e7·23-s + 3.68e8·24-s + 1.30e9·25-s − 3.47e8·27-s + 2.36e9·28-s − 5.19e8·29-s + 1.14e9·30-s + ⋯ |
| L(s) = 1 | − 0.145·2-s + 1.36·3-s − 0.978·4-s − 1.43·5-s − 0.198·6-s − 0.946·7-s + 0.287·8-s + 0.873·9-s + 0.208·10-s − 0.0475·11-s − 1.34·12-s + 0.137·14-s − 1.97·15-s + 0.937·16-s − 1.56·17-s − 0.126·18-s − 0.950·19-s + 1.40·20-s − 1.29·21-s + 0.00691·22-s − 0.0162·23-s + 0.393·24-s + 1.07·25-s − 0.172·27-s + 0.926·28-s − 0.162·29-s + 0.286·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.07985617187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07985617187\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 13.1T + 8.19e3T^{2} \) |
| 3 | \( 1 - 1.72e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 5.02e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.94e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 2.79e5T + 3.45e13T^{2} \) |
| 17 | \( 1 + 1.55e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.94e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.15e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.19e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.34e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.24e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.46e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.43e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.66e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.34e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 2.10e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 7.12e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.14e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 3.36e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 9.46e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.95e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.11e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.33e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.00e12T + 6.73e25T^{2} \) |
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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
−10.12758527092439549235382105834, −9.004880494855973938948388499283, −8.613029403733447993415507210728, −7.72877874142325683247660119382, −6.68480373010734748840406140411, −4.80977589935726667113805092874, −3.76279392644590552121410382676, −3.38670310253756894699870332597, −1.96557398413466195169208275271, −0.10955263139065136794255479498,
0.10955263139065136794255479498, 1.96557398413466195169208275271, 3.38670310253756894699870332597, 3.76279392644590552121410382676, 4.80977589935726667113805092874, 6.68480373010734748840406140411, 7.72877874142325683247660119382, 8.613029403733447993415507210728, 9.004880494855973938948388499283, 10.12758527092439549235382105834
