L(s) = 1 | − 5·2-s + 3·3-s + 17·4-s − 14·5-s − 15·6-s − 32·7-s − 45·8-s + 9·9-s + 70·10-s − 11·11-s + 51·12-s − 38·13-s + 160·14-s − 42·15-s + 89·16-s − 2·17-s − 45·18-s + 72·19-s − 238·20-s − 96·21-s + 55·22-s + 68·23-s − 135·24-s + 71·25-s + 190·26-s + 27·27-s − 544·28-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.25·5-s − 1.02·6-s − 1.72·7-s − 1.98·8-s + 1/3·9-s + 2.21·10-s − 0.301·11-s + 1.22·12-s − 0.810·13-s + 3.05·14-s − 0.722·15-s + 1.39·16-s − 0.0285·17-s − 0.589·18-s + 0.869·19-s − 2.66·20-s − 0.997·21-s + 0.533·22-s + 0.616·23-s − 1.14·24-s + 0.567·25-s + 1.43·26-s + 0.192·27-s − 3.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 - 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 174 T + p^{3} T^{2} \) |
| 41 | \( 1 - 94 T + p^{3} T^{2} \) |
| 43 | \( 1 + 528 T + p^{3} T^{2} \) |
| 47 | \( 1 + 340 T + p^{3} T^{2} \) |
| 53 | \( 1 + 438 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 460 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1092 T + p^{3} T^{2} \) |
| 73 | \( 1 - 562 T + p^{3} T^{2} \) |
| 79 | \( 1 + 16 T + p^{3} T^{2} \) |
| 83 | \( 1 - 372 T + p^{3} T^{2} \) |
| 89 | \( 1 + 966 T + p^{3} T^{2} \) |
| 97 | \( 1 + 526 T + p^{3} T^{2} \) |
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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
−16.01373331971803535604520658707, −15.10326458051728747934703527497, −12.89153469052912144347423721556, −11.56715041911891389839645523290, −10.05912369008524940219976852127, −9.204124258956680265368272925335, −7.83502351517021965034170796576, −6.91749548994513994727963273880, −3.14217182018133821800082250424, 0,
3.14217182018133821800082250424, 6.91749548994513994727963273880, 7.83502351517021965034170796576, 9.204124258956680265368272925335, 10.05912369008524940219976852127, 11.56715041911891389839645523290, 12.89153469052912144347423721556, 15.10326458051728747934703527497, 16.01373331971803535604520658707