| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 13-s − 2·19-s + 2·21-s + 2·23-s − 27-s − 4·29-s + 4·31-s + 4·33-s − 2·37-s + 39-s − 6·41-s − 4·43-s − 8·47-s − 3·49-s − 2·53-s + 2·57-s − 4·59-s + 2·61-s − 2·63-s + 8·67-s − 2·69-s + 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.458·19-s + 0.436·21-s + 0.417·23-s − 0.192·27-s − 0.742·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.264·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s + 0.977·67-s − 0.240·69-s + 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.60220584991910, −13.87323326484416, −13.34161668577308, −13.04589360495607, −12.46844394867404, −12.17359303568295, −11.28837349080273, −11.11261913257933, −10.37399038502896, −9.997623887696467, −9.593070453114954, −8.904886279559688, −8.224336909028022, −7.823476535190584, −7.147165515583276, −6.556761866960768, −6.255931066472610, −5.423782832997473, −5.029048150769359, −4.553520289045575, −3.562589781300158, −3.228034814325526, −2.371121318595042, −1.786597754438423, −0.6742683518139524, 0,
0.6742683518139524, 1.786597754438423, 2.371121318595042, 3.228034814325526, 3.562589781300158, 4.553520289045575, 5.029048150769359, 5.423782832997473, 6.255931066472610, 6.556761866960768, 7.147165515583276, 7.823476535190584, 8.224336909028022, 8.904886279559688, 9.593070453114954, 9.997623887696467, 10.37399038502896, 11.11261913257933, 11.28837349080273, 12.17359303568295, 12.46844394867404, 13.04589360495607, 13.34161668577308, 13.87323326484416, 14.60220584991910
