| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 5·11-s + 13-s − 14-s + 16-s − 17-s − 19-s − 20-s − 5·22-s − 5·23-s − 4·25-s + 26-s − 28-s − 29-s − 6·31-s + 32-s − 34-s + 35-s + 7·37-s − 38-s − 40-s − 2·41-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 − 1.50·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.229·19-s − 0.223·20-s − 1.06·22-s − 1.04·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s − 0.185·29-s − 1.07·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s + 1.15·37-s − 0.162·38-s − 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.951187799408442413225861722767, −7.83195642869312453580393936381, −7.58771647878142615673084205238, −6.34336856013115106006521996723, −5.72816854709225568428895208685, −4.77808523077732043316959668397, −3.91582934000479521066630000171, −3.00954150321325090761034987812, −1.99707427963496596135030374522, 0,
1.99707427963496596135030374522, 3.00954150321325090761034987812, 3.91582934000479521066630000171, 4.77808523077732043316959668397, 5.72816854709225568428895208685, 6.34336856013115106006521996723, 7.58771647878142615673084205238, 7.83195642869312453580393936381, 8.951187799408442413225861722767
