| L(s) = 1 | − 27.8·2-s − 234.·3-s + 263.·4-s − 396.·5-s + 6.52e3·6-s − 1.07e4·7-s + 6.91e3·8-s + 3.51e4·9-s + 1.10e4·10-s + 8.21e4·11-s − 6.17e4·12-s + 6.64e4·13-s + 3.00e5·14-s + 9.28e4·15-s − 3.27e5·16-s − 3.68e5·17-s − 9.79e5·18-s + 2.24e5·19-s − 1.04e5·20-s + 2.52e6·21-s − 2.28e6·22-s + 2.82e5·23-s − 1.61e6·24-s − 1.79e6·25-s − 1.85e6·26-s − 3.62e6·27-s − 2.84e6·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.66·3-s + 0.515·4-s − 0.283·5-s + 2.05·6-s − 1.69·7-s + 0.596·8-s + 1.78·9-s + 0.349·10-s + 1.69·11-s − 0.859·12-s + 0.645·13-s + 2.08·14-s + 0.473·15-s − 1.24·16-s − 1.06·17-s − 2.19·18-s + 0.394·19-s − 0.146·20-s + 2.83·21-s − 2.08·22-s + 0.210·23-s − 0.996·24-s − 0.919·25-s − 0.794·26-s − 1.31·27-s − 0.874·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.87e6T \) |
| good | 2 | \( 1 + 27.8T + 512T^{2} \) |
| 3 | \( 1 + 234.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 396.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.07e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.64e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.24e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.82e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.89e5T + 2.64e13T^{2} \) |
| 41 | \( 1 - 9.64e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.41e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.05e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.34e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.24e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.87e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.94e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.92e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.11e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.91e8T + 7.60e17T^{2} \) |
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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
−13.43965886982392162272344210006, −12.12588178582884494962493448114, −11.13401123083749319977940200480, −9.979488497042427331337781992257, −9.026963177568561978559833811348, −6.92319803463698950225449323773, −6.21817008981739525307714419798, −4.12376238870659213449523118922, −1.04211091021038006701599095220, 0,
1.04211091021038006701599095220, 4.12376238870659213449523118922, 6.21817008981739525307714419798, 6.92319803463698950225449323773, 9.026963177568561978559833811348, 9.979488497042427331337781992257, 11.13401123083749319977940200480, 12.12588178582884494962493448114, 13.43965886982392162272344210006
