| L(s) = 1 | + 2.16·2-s + 2.68·4-s + 0.888·5-s + 1.47·8-s + 1.92·10-s + 3.29·11-s + 13-s − 2.17·16-s + 0.924·17-s + 8.32·19-s + 2.38·20-s + 7.12·22-s − 4.83·23-s − 4.21·25-s + 2.16·26-s − 2.45·29-s − 0.958·31-s − 7.65·32-s + 2·34-s + 10.1·37-s + 18.0·38-s + 1.30·40-s + 5.44·41-s + 4.96·43-s + 8.83·44-s − 10.4·46-s + 9.38·47-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.34·4-s + 0.397·5-s + 0.521·8-s + 0.607·10-s + 0.993·11-s + 0.277·13-s − 0.543·16-s + 0.224·17-s + 1.90·19-s + 0.532·20-s + 1.51·22-s − 1.00·23-s − 0.842·25-s + 0.424·26-s − 0.456·29-s − 0.172·31-s − 1.35·32-s + 0.342·34-s + 1.66·37-s + 2.92·38-s + 0.206·40-s + 0.849·41-s + 0.757·43-s + 1.33·44-s − 1.54·46-s + 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.766134025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.766134025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 5 | \( 1 - 0.888T + 5T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 17 | \( 1 - 0.924T + 17T^{2} \) |
| 19 | \( 1 - 8.32T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 + 0.958T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 - 9.38T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 3.51T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 - 0.561T + 73T^{2} \) |
| 79 | \( 1 + 2.97T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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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
−7.78250869343440827756250744278, −7.27000647515764766817720233170, −6.34735237052507715294239033486, −5.75434222287546457213905018065, −5.42886892206426915743750599490, −4.22411983586180076117502451785, −3.94751686584070549673545482325, −3.01822647778785856757669929820, −2.19275217156569708591712929144, −1.07446848062823242554034962641,
1.07446848062823242554034962641, 2.19275217156569708591712929144, 3.01822647778785856757669929820, 3.94751686584070549673545482325, 4.22411983586180076117502451785, 5.42886892206426915743750599490, 5.75434222287546457213905018065, 6.34735237052507715294239033486, 7.27000647515764766817720233170, 7.78250869343440827756250744278
