| L(s) = 1 | − 2·4-s + 5-s − 7-s − 5·11-s − 5·13-s + 4·16-s − 5·19-s − 2·20-s − 23-s − 4·25-s + 2·28-s − 6·29-s + 6·31-s − 35-s − 4·37-s + 7·41-s − 7·43-s + 10·44-s − 6·47-s + 49-s + 10·52-s − 6·53-s − 5·55-s − 14·59-s − 8·64-s − 5·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s − 1.50·11-s − 1.38·13-s + 16-s − 1.14·19-s − 0.447·20-s − 0.208·23-s − 4/5·25-s + 0.377·28-s − 1.11·29-s + 1.07·31-s − 0.169·35-s − 0.657·37-s + 1.09·41-s − 1.06·43-s + 1.50·44-s − 0.875·47-s + 1/7·49-s + 1.38·52-s − 0.824·53-s − 0.674·55-s − 1.82·59-s − 64-s − 0.620·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.48373929456364, −15.70359960714659, −15.09116615935728, −14.80649224652657, −13.95720186028496, −13.60498014112765, −13.05596883669174, −12.58622490615863, −12.20652327142430, −11.28827160216765, −10.49320946061077, −10.16863439394407, −9.615232591690029, −9.183489940582678, −8.356035466728389, −7.851953000793215, −7.394343411669645, −6.406126279783069, −5.848399755985120, −5.133925944686322, −4.736093368514925, −4.019310051304791, −3.086804939983469, −2.471189082322196, −1.636982236925206, 0, 0,
1.636982236925206, 2.471189082322196, 3.086804939983469, 4.019310051304791, 4.736093368514925, 5.133925944686322, 5.848399755985120, 6.406126279783069, 7.394343411669645, 7.851953000793215, 8.356035466728389, 9.183489940582678, 9.615232591690029, 10.16863439394407, 10.49320946061077, 11.28827160216765, 12.20652327142430, 12.58622490615863, 13.05596883669174, 13.60498014112765, 13.95720186028496, 14.80649224652657, 15.09116615935728, 15.70359960714659, 16.48373929456364
