| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 4·21-s + 25-s − 27-s + 2·29-s + 4·35-s + 10·37-s + 2·39-s − 6·41-s + 43-s − 45-s + 8·47-s + 9·49-s − 2·51-s + 2·53-s + 4·57-s + 2·61-s − 4·63-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.676·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.274·53-s + 0.529·57-s + 0.256·61-s − 0.503·63-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−16.70830735548138, −16.49020082557611, −15.74739960573932, −15.32192397383551, −14.72832047713879, −13.97008582328032, −13.24442607745673, −12.73329936663551, −12.35446906805808, −11.74813751847334, −11.08127697447938, −10.34883510375092, −9.949961619131375, −9.334813931603454, −8.638399391511259, −7.825305180649583, −7.173593114646315, −6.554390929373947, −6.069306726763187, −5.328846067829153, −4.457068043770168, −3.838909539851591, −3.042037416031080, −2.286715961944963, −0.8879695969334523, 0,
0.8879695969334523, 2.286715961944963, 3.042037416031080, 3.838909539851591, 4.457068043770168, 5.328846067829153, 6.069306726763187, 6.554390929373947, 7.173593114646315, 7.825305180649583, 8.638399391511259, 9.334813931603454, 9.949961619131375, 10.34883510375092, 11.08127697447938, 11.74813751847334, 12.35446906805808, 12.73329936663551, 13.24442607745673, 13.97008582328032, 14.72832047713879, 15.32192397383551, 15.74739960573932, 16.49020082557611, 16.70830735548138
