| L(s) = 1 | − 2.58·3-s − 2.81·7-s + 3.69·9-s + 1.58·11-s − 0.888·13-s + 3.98·17-s + 19-s + 7.27·21-s − 3.30·23-s − 1.81·27-s + 6.98·29-s − 4.51·31-s − 4.11·33-s + 2.41·37-s + 2.30·39-s + 5.09·41-s − 8.17·43-s + 9.08·47-s + 0.901·49-s − 10.3·51-s − 7.79·53-s − 2.58·57-s − 2.22·59-s − 9.90·61-s − 10.3·63-s + 7.56·67-s + 8.54·69-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 1.06·7-s + 1.23·9-s + 0.478·11-s − 0.246·13-s + 0.967·17-s + 0.229·19-s + 1.58·21-s − 0.688·23-s − 0.348·27-s + 1.29·29-s − 0.810·31-s − 0.715·33-s + 0.396·37-s + 0.368·39-s + 0.796·41-s − 1.24·43-s + 1.32·47-s + 0.128·49-s − 1.44·51-s − 1.07·53-s − 0.342·57-s − 0.289·59-s − 1.26·61-s − 1.31·63-s + 0.924·67-s + 1.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 + 9.90T + 61T^{2} \) |
| 67 | \( 1 - 7.56T + 67T^{2} \) |
| 71 | \( 1 + 0.777T + 71T^{2} \) |
| 73 | \( 1 - 0.876T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−9.017332873402318905296629812349, −7.85994217001762863168567379537, −6.98150973979863020460703349302, −6.29357708955303672461640857227, −5.77246462337822238244804741764, −4.90639725851899570369214566549, −3.94046108998624592801582996816, −2.89571547232867225181889414913, −1.24315191819665381359586901850, 0,
1.24315191819665381359586901850, 2.89571547232867225181889414913, 3.94046108998624592801582996816, 4.90639725851899570369214566549, 5.77246462337822238244804741764, 6.29357708955303672461640857227, 6.98150973979863020460703349302, 7.85994217001762863168567379537, 9.017332873402318905296629812349
