| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·11-s + 13-s − 16-s − 2·17-s − 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s + 26-s + 2·29-s − 8·31-s + 5·32-s − 2·34-s + 6·37-s − 4·38-s + 3·40-s + 6·41-s − 4·43-s + 4·44-s − 8·46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.986·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.36766611156745383388084599979, −9.359576043548733628235969266343, −8.404296066573383401625737694753, −7.73233535694260719129401432761, −6.37803160504271454727036453417, −5.52200307165618101356281230656, −4.51284738641056261325871513649, −3.73104630189750446831233920392, −2.43381471255309129843962850009, 0,
2.43381471255309129843962850009, 3.73104630189750446831233920392, 4.51284738641056261325871513649, 5.52200307165618101356281230656, 6.37803160504271454727036453417, 7.73233535694260719129401432761, 8.404296066573383401625737694753, 9.359576043548733628235969266343, 10.36766611156745383388084599979
