| L(s) = 1 | + 2-s + 4-s + 4.27·5-s + 7-s + 8-s + 4.27·10-s + 2.27·11-s − 13-s + 14-s + 16-s − 0.274·17-s − 2.27·19-s + 4.27·20-s + 2.27·22-s − 2.27·23-s + 13.2·25-s − 26-s + 28-s − 8.27·29-s + 8·31-s + 32-s − 0.274·34-s + 4.27·35-s − 4.27·37-s − 2.27·38-s + 4.27·40-s − 6.54·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.91·5-s + 0.377·7-s + 0.353·8-s + 1.35·10-s + 0.685·11-s − 0.277·13-s + 0.267·14-s + 0.250·16-s − 0.0666·17-s − 0.521·19-s + 0.955·20-s + 0.485·22-s − 0.474·23-s + 2.65·25-s − 0.196·26-s + 0.188·28-s − 1.53·29-s + 1.43·31-s + 0.176·32-s − 0.0471·34-s + 0.722·35-s − 0.702·37-s − 0.369·38-s + 0.675·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.956987847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.956987847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 17 | \( 1 + 0.274T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + 8.27T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.454131952874410200835377174410, −8.774392513554464436373532761636, −7.65711854166476661794859283000, −6.51158812316118456393107736565, −6.21366204116806777021948303781, −5.26249166449516755712738337369, −4.61878615326246762272471533755, −3.34365699255899355915178313400, −2.19348546702647487436331184116, −1.54817392547638744395084355869,
1.54817392547638744395084355869, 2.19348546702647487436331184116, 3.34365699255899355915178313400, 4.61878615326246762272471533755, 5.26249166449516755712738337369, 6.21366204116806777021948303781, 6.51158812316118456393107736565, 7.65711854166476661794859283000, 8.774392513554464436373532761636, 9.454131952874410200835377174410
