| L(s) = 1 | + 0.625·2-s − 1.60·4-s + 3.29·5-s − 1.24·7-s − 2.25·8-s + 2.05·10-s + 1.56·11-s + 0.814·13-s − 0.776·14-s + 1.80·16-s + 5.32·17-s + 19-s − 5.29·20-s + 0.980·22-s + 0.971·23-s + 5.83·25-s + 0.509·26-s + 1.99·28-s + 7.80·29-s + 2.39·31-s + 5.64·32-s + 3.32·34-s − 4.08·35-s − 2.63·37-s + 0.625·38-s − 7.43·40-s − 11.7·41-s + ⋯ |
| L(s) = 1 | + 0.442·2-s − 0.804·4-s + 1.47·5-s − 0.469·7-s − 0.798·8-s + 0.651·10-s + 0.472·11-s + 0.226·13-s − 0.207·14-s + 0.451·16-s + 1.29·17-s + 0.229·19-s − 1.18·20-s + 0.209·22-s + 0.202·23-s + 1.16·25-s + 0.0999·26-s + 0.377·28-s + 1.45·29-s + 0.429·31-s + 0.997·32-s + 0.570·34-s − 0.690·35-s − 0.432·37-s + 0.101·38-s − 1.17·40-s − 1.83·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.908267352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908267352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.625T + 2T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.814T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 23 | \( 1 - 0.971T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 - 2.99T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.994875742017625900989655848195, −6.80187818091794687043481645989, −6.33460443196041530587901021174, −5.65781360343588926690486978187, −5.15631044400702481318999008796, −4.41084782975267688166128938603, −3.37840576593265569238937767627, −2.93637449360993615273998110845, −1.71831284764609506223416110846, −0.836190975712828585264863857421,
0.836190975712828585264863857421, 1.71831284764609506223416110846, 2.93637449360993615273998110845, 3.37840576593265569238937767627, 4.41084782975267688166128938603, 5.15631044400702481318999008796, 5.65781360343588926690486978187, 6.33460443196041530587901021174, 6.80187818091794687043481645989, 7.994875742017625900989655848195
