| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 20-s + 25-s − 4·26-s − 2·28-s − 8·29-s − 31-s − 32-s + 6·34-s − 2·35-s + 4·37-s − 40-s − 10·41-s + 8·43-s + 4·47-s − 3·49-s − 50-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 1/5·25-s − 0.784·26-s − 0.377·28-s − 1.48·29-s − 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.657·37-s − 0.158·40-s − 1.56·41-s + 1.21·43-s + 0.583·47-s − 3/7·49-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.736880779463300026877543577554, −7.68378196833310505351006793318, −6.91025413756572000512757594118, −6.21881102965997724253631115668, −5.65751449219398047877913070625, −4.37755450809302032813521596801, −3.45847952437465463730423854323, −2.45848603266431580474701776074, −1.46239928898903404461019333123, 0,
1.46239928898903404461019333123, 2.45848603266431580474701776074, 3.45847952437465463730423854323, 4.37755450809302032813521596801, 5.65751449219398047877913070625, 6.21881102965997724253631115668, 6.91025413756572000512757594118, 7.68378196833310505351006793318, 8.736880779463300026877543577554
