| L(s) = 1 | − 2-s + 1.72·3-s + 4-s + 3.59·5-s − 1.72·6-s + 3.50·7-s − 8-s − 0.0372·9-s − 3.59·10-s + 11-s + 1.72·12-s − 5.80·13-s − 3.50·14-s + 6.18·15-s + 16-s + 0.799·17-s + 0.0372·18-s + 3.59·20-s + 6.02·21-s − 22-s + 2.42·23-s − 1.72·24-s + 7.91·25-s + 5.80·26-s − 5.22·27-s + 3.50·28-s + 7.00·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.993·3-s + 0.5·4-s + 1.60·5-s − 0.702·6-s + 1.32·7-s − 0.353·8-s − 0.0124·9-s − 1.13·10-s + 0.301·11-s + 0.496·12-s − 1.60·13-s − 0.935·14-s + 1.59·15-s + 0.250·16-s + 0.193·17-s + 0.00879·18-s + 0.803·20-s + 1.31·21-s − 0.213·22-s + 0.505·23-s − 0.351·24-s + 1.58·25-s + 1.13·26-s − 1.00·27-s + 0.661·28-s + 1.30·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.456229651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.456229651\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 - 0.799T + 17T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 0.227T + 71T^{2} \) |
| 73 | \( 1 + 9.26T + 73T^{2} \) |
| 79 | \( 1 + 0.915T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 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
−7.998035280289755451906468684177, −7.33223107512466063996885114851, −6.65438574092096762015269893501, −5.70049592919098381629519013138, −5.17143100235242114941625122940, −4.39026799820750100843246177721, −3.06463802671642658807663564033, −2.34839751320166885969799978532, −1.98918890054352541304197820971, −1.01484868742488539526934553199,
1.01484868742488539526934553199, 1.98918890054352541304197820971, 2.34839751320166885969799978532, 3.06463802671642658807663564033, 4.39026799820750100843246177721, 5.17143100235242114941625122940, 5.70049592919098381629519013138, 6.65438574092096762015269893501, 7.33223107512466063996885114851, 7.998035280289755451906468684177
