L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s − 11-s − 6·13-s − 14-s + 16-s + 5·17-s − 7·19-s − 2·20-s − 22-s − 4·23-s − 25-s − 6·26-s − 28-s + 4·29-s − 6·31-s + 32-s + 5·34-s + 2·35-s + 2·37-s − 7·38-s − 2·40-s − 3·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 1.60·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.742·29-s − 1.07·31-s + 0.176·32-s + 0.857·34-s + 0.338·35-s + 0.328·37-s − 1.13·38-s − 0.316·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.624236591226405116055534482229, −8.299989394292870205760058751944, −7.68990678626982471706890997877, −6.91864926329982790755048142082, −5.94426472351306417627738918724, −4.95666001421823337682643454787, −4.15236351510117638725696143570, −3.22509762090529529094711071941, −2.14237304692120266544001683935, 0,
2.14237304692120266544001683935, 3.22509762090529529094711071941, 4.15236351510117638725696143570, 4.95666001421823337682643454787, 5.94426472351306417627738918724, 6.91864926329982790755048142082, 7.68990678626982471706890997877, 8.299989394292870205760058751944, 9.624236591226405116055534482229