| L(s) = 1 | + 1.25·2-s + 8.41·3-s − 126.·4-s − 307.·5-s + 10.5·6-s + 83.6·7-s − 319.·8-s − 2.11e3·9-s − 386.·10-s − 3.72e3·11-s − 1.06e3·12-s + 104.·14-s − 2.58e3·15-s + 1.57e4·16-s − 3.55e4·17-s − 2.65e3·18-s − 3.55e4·19-s + 3.89e4·20-s + 704.·21-s − 4.67e3·22-s − 8.46e3·23-s − 2.68e3·24-s + 1.66e4·25-s − 3.62e4·27-s − 1.05e4·28-s + 3.79e4·29-s − 3.24e3·30-s + ⋯ |
| L(s) = 1 | + 0.110·2-s + 0.179·3-s − 0.987·4-s − 1.10·5-s + 0.0199·6-s + 0.0922·7-s − 0.220·8-s − 0.967·9-s − 0.122·10-s − 0.844·11-s − 0.177·12-s + 0.0102·14-s − 0.198·15-s + 0.963·16-s − 1.75·17-s − 0.107·18-s − 1.18·19-s + 1.08·20-s + 0.0165·21-s − 0.0936·22-s − 0.145·23-s − 0.0396·24-s + 0.212·25-s − 0.354·27-s − 0.0910·28-s + 0.288·29-s − 0.0219·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2930539226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2930539226\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 1.25T + 128T^{2} \) |
| 3 | \( 1 - 8.41T + 2.18e3T^{2} \) |
| 5 | \( 1 + 307.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 83.6T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.72e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.55e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.46e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.79e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.61e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.67e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.07e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.31e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.83e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.65e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.86e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.15e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.65e7T + 8.07e13T^{2} \) |
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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
−11.41478528513093313030063146841, −10.65088095839415587110634092060, −9.130446286143450649051427483906, −8.460245649278955637602432235854, −7.62726598722437334988986956161, −6.03604027779909309614068615006, −4.71862621112451349963405042075, −3.89306930155436794514874860537, −2.51559997479836930544384923181, −0.27648539043695843602849712715,
0.27648539043695843602849712715, 2.51559997479836930544384923181, 3.89306930155436794514874860537, 4.71862621112451349963405042075, 6.03604027779909309614068615006, 7.62726598722437334988986956161, 8.460245649278955637602432235854, 9.130446286143450649051427483906, 10.65088095839415587110634092060, 11.41478528513093313030063146841
