| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s + 2·13-s + 16-s + 4·17-s − 18-s + 6·22-s − 2·23-s − 24-s − 2·26-s + 27-s − 8·29-s + 31-s − 32-s − 6·33-s − 4·34-s + 36-s + 6·37-s + 2·39-s − 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.27·22-s − 0.417·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 0.176·32-s − 1.04·33-s − 0.685·34-s + 1/6·36-s + 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.003142688326185229192189291854, −7.57145199347852337456467634428, −6.70125330000359150399512666640, −5.71965374146475163901411770216, −5.17884530172546960029716670442, −3.98192071698501959157320518507, −3.11819571378905520704365893613, −2.40307569824514192694020651257, −1.40232919259954398189652859894, 0,
1.40232919259954398189652859894, 2.40307569824514192694020651257, 3.11819571378905520704365893613, 3.98192071698501959157320518507, 5.17884530172546960029716670442, 5.71965374146475163901411770216, 6.70125330000359150399512666640, 7.57145199347852337456467634428, 8.003142688326185229192189291854
