| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 3·11-s − 14-s + 16-s + 17-s + 4·19-s + 20-s + 3·22-s − 7·23-s + 25-s − 28-s − 2·29-s + 2·31-s + 32-s + 34-s − 35-s − 6·37-s + 4·38-s + 40-s − 2·41-s − 5·43-s + 3·44-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 + 0.904·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.223·20-s + 0.639·22-s − 1.45·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s − 0.986·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-s − 0.762·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.02910543243743, −12.63163397510438, −12.06194199593870, −11.82051565630620, −11.42880976820818, −10.69977880637739, −10.34311009268069, −9.745975828148516, −9.506513792100126, −8.960688336944465, −8.305539290159461, −7.843938677192568, −7.342727866270074, −6.690741706024389, −6.386735619658200, −5.972007967638371, −5.368478780744669, −4.947275335500151, −4.350330070342806, −3.752134056347642, −3.309255135199833, −2.914048088317827, −1.896178392068524, −1.771309960058500, −0.9280913678134339, 0,
0.9280913678134339, 1.771309960058500, 1.896178392068524, 2.914048088317827, 3.309255135199833, 3.752134056347642, 4.350330070342806, 4.947275335500151, 5.368478780744669, 5.972007967638371, 6.386735619658200, 6.690741706024389, 7.342727866270074, 7.843938677192568, 8.305539290159461, 8.960688336944465, 9.506513792100126, 9.745975828148516, 10.34311009268069, 10.69977880637739, 11.42880976820818, 11.82051565630620, 12.06194199593870, 12.63163397510438, 13.02910543243743
