| L(s) = 1 | + 2·3-s − 5-s + 9-s + 3·11-s − 2·13-s − 2·15-s − 2·17-s + 19-s + 23-s − 4·25-s − 4·27-s − 8·29-s + 6·33-s + 2·37-s − 4·39-s + 10·41-s − 11·43-s − 45-s − 3·47-s − 4·51-s − 3·55-s + 2·57-s − 8·59-s + 11·61-s + 2·65-s − 4·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.208·23-s − 4/5·25-s − 0.769·27-s − 1.48·29-s + 1.04·33-s + 0.328·37-s − 0.640·39-s + 1.56·41-s − 1.67·43-s − 0.149·45-s − 0.437·47-s − 0.560·51-s − 0.404·55-s + 0.264·57-s − 1.04·59-s + 1.40·61-s + 0.248·65-s − 0.488·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.63129220464649194784575112128, −7.10257914488936398809686373285, −6.23268374988125788494985607886, −5.43952874116149442291448210562, −4.42554434452564705293866454506, −3.84722183671507879500828306061, −3.17543824678402255630377197222, −2.33092681474701224556298111866, −1.51783528825629656970822017401, 0,
1.51783528825629656970822017401, 2.33092681474701224556298111866, 3.17543824678402255630377197222, 3.84722183671507879500828306061, 4.42554434452564705293866454506, 5.43952874116149442291448210562, 6.23268374988125788494985607886, 7.10257914488936398809686373285, 7.63129220464649194784575112128
