| L(s) = 1 | − 2-s + 4-s + 5-s + 4.50·7-s − 8-s − 10-s − 4.33·11-s − 3.72·13-s − 4.50·14-s + 16-s − 1.11·17-s + 4.50·19-s + 20-s + 4.33·22-s + 23-s + 25-s + 3.72·26-s + 4.50·28-s + 8.23·29-s + 1.72·31-s − 32-s + 1.11·34-s + 4.50·35-s − 0.781·37-s − 4.50·38-s − 40-s − 3.90·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.70·7-s − 0.353·8-s − 0.316·10-s − 1.30·11-s − 1.03·13-s − 1.20·14-s + 0.250·16-s − 0.271·17-s + 1.03·19-s + 0.223·20-s + 0.924·22-s + 0.208·23-s + 0.200·25-s + 0.731·26-s + 0.852·28-s + 1.52·29-s + 0.310·31-s − 0.176·32-s + 0.191·34-s + 0.762·35-s − 0.128·37-s − 0.731·38-s − 0.158·40-s − 0.609·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.562131443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562131443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 0.781T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 + 0.642T + 97T^{2} \) |
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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
−8.938463965076514962842120087198, −8.405226139802127551471903111546, −7.53161551185861690236774491543, −7.23960164687173150727319771864, −5.83773592149012265885980781220, −5.13544407037646207865830470036, −4.51284679004621206678298880350, −2.80595442905507045445129546824, −2.15204519825300887911007438662, −0.946701776765772005968830684738,
0.946701776765772005968830684738, 2.15204519825300887911007438662, 2.80595442905507045445129546824, 4.51284679004621206678298880350, 5.13544407037646207865830470036, 5.83773592149012265885980781220, 7.23960164687173150727319771864, 7.53161551185861690236774491543, 8.405226139802127551471903111546, 8.938463965076514962842120087198
