| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 2·11-s + 13-s + 14-s + 16-s + 3·17-s + 20-s + 2·22-s + 4·23-s − 4·25-s + 26-s + 28-s + 2·29-s − 4·31-s + 32-s + 3·34-s + 35-s − 3·37-s + 40-s − 5·43-s + 2·44-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.223·20-s + 0.426·22-s + 0.834·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.169·35-s − 0.493·37-s + 0.158·40-s − 0.762·43-s + 0.301·44-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84474 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84474 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−14.23270170871939, −13.62729067962968, −13.28698266669658, −12.81711334086098, −12.14257661083263, −11.78333549570029, −11.37320757618557, −10.71394849373289, −10.33820977393198, −9.678790460367650, −9.243585803173620, −8.644564548306784, −8.004134359405401, −7.598072361857074, −6.801171845172943, −6.549904526241087, −5.800747584185348, −5.425615907609626, −4.825901175634098, −4.294222179987117, −3.540309772856769, −3.185690348587705, −2.391902063613329, −1.541588409149582, −1.323443490626808, 0,
1.323443490626808, 1.541588409149582, 2.391902063613329, 3.185690348587705, 3.540309772856769, 4.294222179987117, 4.825901175634098, 5.425615907609626, 5.800747584185348, 6.549904526241087, 6.801171845172943, 7.598072361857074, 8.004134359405401, 8.644564548306784, 9.243585803173620, 9.678790460367650, 10.33820977393198, 10.71394849373289, 11.37320757618557, 11.78333549570029, 12.14257661083263, 12.81711334086098, 13.28698266669658, 13.62729067962968, 14.23270170871939
