| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 2.47·13-s + 4·14-s + 16-s − 2.47·17-s − 18-s + 2·19-s + 4·21-s + 23-s + 24-s − 2.47·26-s − 27-s − 4·28-s − 0.472·29-s − 32-s + 2.47·34-s + 36-s − 0.472·37-s − 2·38-s − 2.47·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 0.333·9-s − 0.288·12-s + 0.685·13-s + 1.06·14-s + 0.250·16-s − 0.599·17-s − 0.235·18-s + 0.458·19-s + 0.872·21-s + 0.208·23-s + 0.204·24-s − 0.484·26-s − 0.192·27-s − 0.755·28-s − 0.0876·29-s − 0.176·32-s + 0.423·34-s + 0.166·36-s − 0.0776·37-s − 0.324·38-s − 0.395·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.290897819595579996332395679537, −7.36869008281921843464669680016, −6.71335103195816184794160801477, −6.15500126483564731020447213590, −5.47847462040112577797905381602, −4.25299530624176670939637830965, −3.38688293818164858612222360780, −2.49380533438491056427850453873, −1.12703442207323600734208240812, 0,
1.12703442207323600734208240812, 2.49380533438491056427850453873, 3.38688293818164858612222360780, 4.25299530624176670939637830965, 5.47847462040112577797905381602, 6.15500126483564731020447213590, 6.71335103195816184794160801477, 7.36869008281921843464669680016, 8.290897819595579996332395679537
