| L(s) = 1 | + 2-s + 4-s + 5-s − 3·7-s + 8-s − 3·9-s + 10-s − 3·11-s − 3·14-s + 16-s − 4·17-s − 3·18-s + 7·19-s + 20-s − 3·22-s − 4·23-s + 25-s − 3·28-s − 8·29-s − 10·31-s + 32-s − 4·34-s − 3·35-s − 3·36-s + 3·37-s + 7·38-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s + 0.353·8-s − 9-s + 0.316·10-s − 0.904·11-s − 0.801·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 1.60·19-s + 0.223·20-s − 0.639·22-s − 0.834·23-s + 1/5·25-s − 0.566·28-s − 1.48·29-s − 1.79·31-s + 0.176·32-s − 0.685·34-s − 0.507·35-s − 1/2·36-s + 0.493·37-s + 1.13·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.300297501624975978492292037818, −7.975452881208269696892965733202, −7.27638050876996681243276873676, −6.25798222389869796675575975368, −5.70991159044996456379942019233, −5.01690284418133212863771660578, −3.67474705141338769051628639672, −2.99754177050974057021226379799, −2.05679095411134817247991307939, 0,
2.05679095411134817247991307939, 2.99754177050974057021226379799, 3.67474705141338769051628639672, 5.01690284418133212863771660578, 5.70991159044996456379942019233, 6.25798222389869796675575975368, 7.27638050876996681243276873676, 7.975452881208269696892965733202, 9.300297501624975978492292037818
