| L(s) = 1 | + 4.76·2-s − 3·3-s + 14.7·4-s + 18.8·5-s − 14.2·6-s − 23.8·7-s + 32.0·8-s + 9·9-s + 89.7·10-s + 60.2·11-s − 44.1·12-s − 113.·14-s − 56.5·15-s + 34.9·16-s − 1.17·17-s + 42.8·18-s + 29.9·19-s + 277.·20-s + 71.5·21-s + 287.·22-s + 159.·23-s − 96.0·24-s + 229.·25-s − 27·27-s − 351.·28-s + 20.8·29-s − 269.·30-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s + 1.68·5-s − 0.972·6-s − 1.28·7-s + 1.41·8-s + 0.333·9-s + 2.83·10-s + 1.65·11-s − 1.06·12-s − 2.17·14-s − 0.972·15-s + 0.545·16-s − 0.0167·17-s + 0.561·18-s + 0.362·19-s + 3.10·20-s + 0.743·21-s + 2.78·22-s + 1.44·23-s − 0.817·24-s + 1.83·25-s − 0.192·27-s − 2.37·28-s + 0.133·29-s − 1.63·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.757976165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.757976165\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.76T + 8T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 + 23.8T + 343T^{2} \) |
| 11 | \( 1 - 60.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 1.17T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 520.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 409.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 74.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 305.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 318.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 867.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 626.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 491.T + 9.12e5T^{2} \) |
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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
−10.72193964618105561373545006555, −9.637377170678571859720174575737, −9.179899381395428436205789953829, −6.77003968694737012386725271435, −6.60523773683204511715707931393, −5.79115444577090241260973025829, −5.00524018136703749004139100679, −3.75764532448328809261617092098, −2.75174472206153401029179723825, −1.38731984103624258405838410421,
1.38731984103624258405838410421, 2.75174472206153401029179723825, 3.75764532448328809261617092098, 5.00524018136703749004139100679, 5.79115444577090241260973025829, 6.60523773683204511715707931393, 6.77003968694737012386725271435, 9.179899381395428436205789953829, 9.637377170678571859720174575737, 10.72193964618105561373545006555
