L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s − 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s − 2·45-s + 2·51-s + 2·53-s + 8·55-s + 4·57-s + 4·59-s − 2·61-s + 4·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 0.280·51-s + 0.274·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.22685430203741355342227556898, −6.28266974817041814701199375242, −5.59620622280165428169903914653, −5.00670845718665013316471514453, −4.12187187931469968184698878478, −3.71533601983866002659839530844, −2.50123083621432021396021244496, −1.80970977960128750436372604769, 0, 0,
1.80970977960128750436372604769, 2.50123083621432021396021244496, 3.71533601983866002659839530844, 4.12187187931469968184698878478, 5.00670845718665013316471514453, 5.59620622280165428169903914653, 6.28266974817041814701199375242, 7.22685430203741355342227556898