| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s − 6·11-s + 2·12-s + 2·13-s + 14-s + 16-s + 6·17-s − 18-s − 8·19-s − 2·21-s + 6·22-s + 23-s − 2·24-s − 2·26-s − 4·27-s − 28-s + 29-s − 6·31-s − 32-s − 12·33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.436·21-s + 1.27·22-s + 0.208·23-s − 0.408·24-s − 0.392·26-s − 0.769·27-s − 0.188·28-s + 0.185·29-s − 1.07·31-s − 0.176·32-s − 2.08·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.12959910054511, −12.80333322039229, −12.37083176898492, −11.72472586448282, −10.92292583394819, −10.85909889337164, −10.21274790365309, −9.915080832956760, −9.308729700790005, −8.899759573689665, −8.418072338778585, −7.939838552467712, −7.809554337395708, −7.230892220884876, −6.574251897434805, −5.969981403988142, −5.599486728655614, −4.978844936491482, −4.225975150353537, −3.601033214924437, −3.166704988308788, −2.616208472544810, −2.231481565812437, −1.621231865927363, −0.7066505531174591, 0,
0.7066505531174591, 1.621231865927363, 2.231481565812437, 2.616208472544810, 3.166704988308788, 3.601033214924437, 4.225975150353537, 4.978844936491482, 5.599486728655614, 5.969981403988142, 6.574251897434805, 7.230892220884876, 7.809554337395708, 7.939838552467712, 8.418072338778585, 8.899759573689665, 9.308729700790005, 9.915080832956760, 10.21274790365309, 10.85909889337164, 10.92292583394819, 11.72472586448282, 12.37083176898492, 12.80333322039229, 13.12959910054511
