| L(s) = 1 | + 2.70·2-s + 0.960·3-s + 5.32·4-s + 3.57·5-s + 2.59·6-s − 3.95·7-s + 8.99·8-s − 2.07·9-s + 9.66·10-s − 11-s + 5.11·12-s − 10.7·14-s + 3.43·15-s + 13.6·16-s + 2.79·17-s − 5.62·18-s + 0.00685·19-s + 19.0·20-s − 3.79·21-s − 2.70·22-s − 3.94·23-s + 8.63·24-s + 7.76·25-s − 4.87·27-s − 21.0·28-s + 6.09·29-s + 9.28·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.554·3-s + 2.66·4-s + 1.59·5-s + 1.06·6-s − 1.49·7-s + 3.17·8-s − 0.692·9-s + 3.05·10-s − 0.301·11-s + 1.47·12-s − 2.86·14-s + 0.885·15-s + 3.42·16-s + 0.678·17-s − 1.32·18-s + 0.00157·19-s + 4.25·20-s − 0.828·21-s − 0.576·22-s − 0.822·23-s + 1.76·24-s + 1.55·25-s − 0.938·27-s − 3.97·28-s + 1.13·29-s + 1.69·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.525009780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.525009780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.960T + 3T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 - 0.00685T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 + 0.242T + 31T^{2} \) |
| 37 | \( 1 + 0.801T + 37T^{2} \) |
| 41 | \( 1 - 6.58T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 - 9.02T + 53T^{2} \) |
| 59 | \( 1 + 4.00T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 + 0.166T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 + 0.547T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 8.78T + 97T^{2} \) |
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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
−9.518973301619075393634408851053, −8.379284513059002854531864481895, −7.21042946660462663807473688076, −6.39436902546612798338535316257, −5.89195947133953518748741094394, −5.41790398359387073740817473749, −4.23915681023374013231739105147, −3.01447808882951793963357447225, −2.89543048127897555257192748223, −1.80074673188425925656886414381,
1.80074673188425925656886414381, 2.89543048127897555257192748223, 3.01447808882951793963357447225, 4.23915681023374013231739105147, 5.41790398359387073740817473749, 5.89195947133953518748741094394, 6.39436902546612798338535316257, 7.21042946660462663807473688076, 8.379284513059002854531864481895, 9.518973301619075393634408851053
