L(s) = 1 | − 3-s − 4·5-s − 7-s − 2·9-s − 2·13-s + 4·15-s − 5·17-s + 7·19-s + 21-s + 11·25-s + 5·27-s − 8·29-s + 2·31-s + 4·35-s + 8·37-s + 2·39-s − 2·41-s + 3·43-s + 8·45-s + 49-s + 5·51-s − 6·53-s − 7·57-s − 9·59-s + 6·61-s + 2·63-s + 8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s − 2/3·9-s − 0.554·13-s + 1.03·15-s − 1.21·17-s + 1.60·19-s + 0.218·21-s + 11/5·25-s + 0.962·27-s − 1.48·29-s + 0.359·31-s + 0.676·35-s + 1.31·37-s + 0.320·39-s − 0.312·41-s + 0.457·43-s + 1.19·45-s + 1/7·49-s + 0.700·51-s − 0.824·53-s − 0.927·57-s − 1.17·59-s + 0.768·61-s + 0.251·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 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
−12.98170281297260, −12.53670892082551, −12.03671226654489, −11.75524390154001, −11.34166520736149, −10.91191692155566, −10.71831471823167, −9.773964108527602, −9.328284778598541, −9.040186617435747, −8.231289899996298, −7.949869688623839, −7.502393984615279, −6.998353363237901, −6.560044154091428, −5.976492161721512, −5.347852107301287, −4.891006491431573, −4.434005620707213, −3.813148742330429, −3.379601606708579, −2.825756389712174, −2.276049985126957, −1.190053981617121, −0.5187472618384308, 0,
0.5187472618384308, 1.190053981617121, 2.276049985126957, 2.825756389712174, 3.379601606708579, 3.813148742330429, 4.434005620707213, 4.891006491431573, 5.347852107301287, 5.976492161721512, 6.560044154091428, 6.998353363237901, 7.502393984615279, 7.949869688623839, 8.231289899996298, 9.040186617435747, 9.328284778598541, 9.773964108527602, 10.71831471823167, 10.91191692155566, 11.34166520736149, 11.75524390154001, 12.03671226654489, 12.53670892082551, 12.98170281297260