| L(s) = 1 | − 2·5-s − 4·7-s − 3·9-s − 2·13-s + 2·17-s + 2·19-s − 6·23-s − 25-s + 6·29-s − 2·31-s + 8·35-s + 37-s − 2·41-s + 2·43-s + 6·45-s − 4·47-s + 9·49-s + 6·53-s − 6·59-s + 6·61-s + 12·63-s + 4·65-s + 8·67-s − 4·71-s + 14·73-s + 2·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 9-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s − 1/5·25-s + 1.11·29-s − 0.359·31-s + 1.35·35-s + 0.164·37-s − 0.312·41-s + 0.304·43-s + 0.894·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 1.51·63-s + 0.496·65-s + 0.977·67-s − 0.474·71-s + 1.63·73-s + 0.225·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6379847301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6379847301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.973965603903636884920497152678, −8.139963688567899948562200386516, −7.53219516094291790010148313731, −6.59864521555049293791212644521, −5.97021818307546082272950549982, −5.06427273115980897713846035422, −3.87796534605450220035734187243, −3.30488835115774555617471587930, −2.42538163944277566108766365040, −0.48051744224849779476855570293,
0.48051744224849779476855570293, 2.42538163944277566108766365040, 3.30488835115774555617471587930, 3.87796534605450220035734187243, 5.06427273115980897713846035422, 5.97021818307546082272950549982, 6.59864521555049293791212644521, 7.53219516094291790010148313731, 8.139963688567899948562200386516, 8.973965603903636884920497152678
