| L(s) = 1 | − 2-s + 4-s + 0.482·5-s − 1.51·7-s − 8-s − 0.482·10-s + 6.21·11-s + 0.831·13-s + 1.51·14-s + 16-s − 1.73·17-s + 5.08·19-s + 0.482·20-s − 6.21·22-s + 1.86·23-s − 4.76·25-s − 0.831·26-s − 1.51·28-s + 4.51·29-s − 1.13·31-s − 32-s + 1.73·34-s − 0.732·35-s − 4.04·37-s − 5.08·38-s − 0.482·40-s + 41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.215·5-s − 0.573·7-s − 0.353·8-s − 0.152·10-s + 1.87·11-s + 0.230·13-s + 0.405·14-s + 0.250·16-s − 0.420·17-s + 1.16·19-s + 0.107·20-s − 1.32·22-s + 0.389·23-s − 0.953·25-s − 0.163·26-s − 0.286·28-s + 0.838·29-s − 0.203·31-s − 0.176·32-s + 0.297·34-s − 0.123·35-s − 0.665·37-s − 0.824·38-s − 0.0763·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412099782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412099782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 0.482T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 6.21T + 11T^{2} \) |
| 13 | \( 1 - 0.831T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 - 3.73T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 9.41T + 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 - 1.55T + 79T^{2} \) |
| 83 | \( 1 - 4.41T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 9.54T + 97T^{2} \) |
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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
−9.151559995689987986034645220873, −8.484986913626507083103337133373, −7.47009103733907815344257589561, −6.67699951307830881250874247535, −6.25402182161133559874129876711, −5.19367150422404047551296888596, −3.95149049107055907013847255030, −3.23924260154561909629782948583, −1.93430577336928335148625093198, −0.898999575744076584925870761664,
0.898999575744076584925870761664, 1.93430577336928335148625093198, 3.23924260154561909629782948583, 3.95149049107055907013847255030, 5.19367150422404047551296888596, 6.25402182161133559874129876711, 6.67699951307830881250874247535, 7.47009103733907815344257589561, 8.484986913626507083103337133373, 9.151559995689987986034645220873
