| L(s) = 1 | − 1.74·2-s + 1.04·4-s + 3.62·5-s + 1.82·7-s + 1.66·8-s − 6.33·10-s − 0.911·11-s + 0.527·13-s − 3.18·14-s − 4.99·16-s + 4.49·17-s − 19-s + 3.80·20-s + 1.59·22-s − 3.21·23-s + 8.14·25-s − 0.921·26-s + 1.91·28-s + 1.72·29-s + 10.5·31-s + 5.40·32-s − 7.85·34-s + 6.60·35-s + 2.04·37-s + 1.74·38-s + 6.02·40-s + 2.59·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.524·4-s + 1.62·5-s + 0.688·7-s + 0.587·8-s − 2.00·10-s − 0.274·11-s + 0.146·13-s − 0.850·14-s − 1.24·16-s + 1.09·17-s − 0.229·19-s + 0.850·20-s + 0.339·22-s − 0.670·23-s + 1.62·25-s − 0.180·26-s + 0.361·28-s + 0.320·29-s + 1.89·31-s + 0.955·32-s − 1.34·34-s + 1.11·35-s + 0.336·37-s + 0.283·38-s + 0.952·40-s + 0.404·41-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
| \(L(1)\) |
\(\approx\) |
\(1.709873879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709873879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.74T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 0.911T + 11T^{2} \) |
| 13 | \( 1 - 0.527T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 8.49T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.036229743319520553289457526211, −7.41526948509664264472332721224, −6.32842518147907674059821978932, −6.06125987307124805653885083960, −4.97580026841333876517618189700, −4.62018857666114075678540936271, −3.20659067009699246807685373185, −2.23402762895703961452621254828, −1.61462485803902871651534101581, −0.841696143236553853944754324681,
0.841696143236553853944754324681, 1.61462485803902871651534101581, 2.23402762895703961452621254828, 3.20659067009699246807685373185, 4.62018857666114075678540936271, 4.97580026841333876517618189700, 6.06125987307124805653885083960, 6.32842518147907674059821978932, 7.41526948509664264472332721224, 8.036229743319520553289457526211
