L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·11-s − 2·15-s + 2·17-s + 2·23-s − 25-s + 27-s − 6·29-s − 4·31-s + 2·33-s − 6·37-s − 2·41-s − 2·45-s + 2·51-s + 6·53-s − 4·55-s − 12·59-s − 12·61-s + 12·67-s + 2·69-s − 10·71-s + 12·73-s − 75-s + 12·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.516·15-s + 0.485·17-s + 0.417·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.312·41-s − 0.298·45-s + 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.56·59-s − 1.53·61-s + 1.46·67-s + 0.240·69-s − 1.18·71-s + 1.40·73-s − 0.115·75-s + 1.35·79-s + 1/9·81-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.42309020386561355734136044872, −6.91879740136762588622017109624, −6.02896285682546558556644118498, −5.22963397377530505532416027174, −4.41453266601339998935853948194, −3.65708983365160016400226497042, −3.31612689995187390461845623381, −2.17608545931971488627920918893, −1.29412681328680920762964687585, 0,
1.29412681328680920762964687585, 2.17608545931971488627920918893, 3.31612689995187390461845623381, 3.65708983365160016400226497042, 4.41453266601339998935853948194, 5.22963397377530505532416027174, 6.02896285682546558556644118498, 6.91879740136762588622017109624, 7.42309020386561355734136044872