| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 13-s + 2·15-s + 16-s − 17-s + 18-s + 8·19-s − 2·20-s + 22-s − 4·23-s − 24-s − 25-s + 26-s − 27-s + 6·29-s + 2·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303668043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303668043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.89263527343359, −15.72400527982683, −15.12743662819552, −14.17900147858361, −14.01239324739613, −13.34014799623573, −12.53901534211780, −12.00388759571822, −11.85791684727675, −11.10517542820850, −10.69832079610569, −9.874954567895536, −9.338705940202399, −8.401089183498101, −7.862985956970226, −7.170630739738205, −6.763904703641339, −5.874326071804082, −5.410472558636607, −4.699927905141964, −3.936910767099510, −3.560927908968328, −2.643227960954291, −1.574967149751973, −0.6356293898749369,
0.6356293898749369, 1.574967149751973, 2.643227960954291, 3.560927908968328, 3.936910767099510, 4.699927905141964, 5.410472558636607, 5.874326071804082, 6.763904703641339, 7.170630739738205, 7.862985956970226, 8.401089183498101, 9.338705940202399, 9.874954567895536, 10.69832079610569, 11.10517542820850, 11.85791684727675, 12.00388759571822, 12.53901534211780, 13.34014799623573, 14.01239324739613, 14.17900147858361, 15.12743662819552, 15.72400527982683, 15.89263527343359
