| L(s) = 1 | − 5-s + 5·11-s + 13-s − 5·17-s − 4·19-s + 3·23-s + 25-s + 8·29-s + 9·31-s + 8·37-s + 5·41-s + 43-s + 8·47-s − 7·49-s + 13·53-s − 5·55-s + 8·59-s − 8·61-s − 65-s − 3·67-s − 8·71-s + 4·73-s − 9·83-s + 5·85-s − 12·89-s + 4·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s + 0.277·13-s − 1.21·17-s − 0.917·19-s + 0.625·23-s + 1/5·25-s + 1.48·29-s + 1.61·31-s + 1.31·37-s + 0.780·41-s + 0.152·43-s + 1.16·47-s − 49-s + 1.78·53-s − 0.674·55-s + 1.04·59-s − 1.02·61-s − 0.124·65-s − 0.366·67-s − 0.949·71-s + 0.468·73-s − 0.987·83-s + 0.542·85-s − 1.27·89-s + 0.410·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524498925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524498925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−15.05179926578867, −14.62768222633607, −14.11750410883171, −13.42708295488683, −13.11405441175383, −12.27057667309507, −11.95648970364914, −11.38879220439335, −10.90799424651124, −10.35061653312198, −9.641470020923263, −9.054878394864516, −8.561482479950734, −8.224039749289192, −7.277734381078107, −6.763526108558472, −6.334534188752602, −5.802600346861575, −4.621545391628953, −4.395507044785718, −3.878695745201328, −2.893555645259988, −2.372434084283553, −1.290802306965514, −0.6760609821304975,
0.6760609821304975, 1.290802306965514, 2.372434084283553, 2.893555645259988, 3.878695745201328, 4.395507044785718, 4.621545391628953, 5.802600346861575, 6.334534188752602, 6.763526108558472, 7.277734381078107, 8.224039749289192, 8.561482479950734, 9.054878394864516, 9.641470020923263, 10.35061653312198, 10.90799424651124, 11.38879220439335, 11.95648970364914, 12.27057667309507, 13.11405441175383, 13.42708295488683, 14.11750410883171, 14.62768222633607, 15.05179926578867
