| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 5-s + 4.82·7-s + 4.41·8-s − 2.41·10-s − 3.41·11-s − 13-s + 11.6·14-s + 2.99·16-s − 0.828·17-s + 0.585·19-s − 3.82·20-s − 8.24·22-s − 1.41·23-s + 25-s − 2.41·26-s + 18.4·28-s + 5.65·29-s + 1.75·31-s − 1.58·32-s − 1.99·34-s − 4.82·35-s − 8.48·37-s + 1.41·38-s − 4.41·40-s + 3.17·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.447·5-s + 1.82·7-s + 1.56·8-s − 0.763·10-s − 1.02·11-s − 0.277·13-s + 3.11·14-s + 0.749·16-s − 0.200·17-s + 0.134·19-s − 0.856·20-s − 1.75·22-s − 0.294·23-s + 0.200·25-s − 0.473·26-s + 3.49·28-s + 1.05·29-s + 0.315·31-s − 0.280·32-s − 0.342·34-s − 0.816·35-s − 1.39·37-s + 0.229·38-s − 0.697·40-s + 0.495·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.876629761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.876629761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−11.02327904860097487853365600695, −10.36110934961180151067116332977, −8.601144137217098123137355612783, −7.82748707143792572786159821091, −7.00274043662204370004099982388, −5.70839080855422604232435173677, −4.89361651758623846143271627664, −4.42392998264889679461814329749, −3.07921870702749460485365586304, −1.91204956050814003374058696388,
1.91204956050814003374058696388, 3.07921870702749460485365586304, 4.42392998264889679461814329749, 4.89361651758623846143271627664, 5.70839080855422604232435173677, 7.00274043662204370004099982388, 7.82748707143792572786159821091, 8.601144137217098123137355612783, 10.36110934961180151067116332977, 11.02327904860097487853365600695
