| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 3·17-s − 18-s − 20-s + 21-s − 22-s + 8·23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155298 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155298 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 181 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.52448538872724, −12.86337949030930, −12.50162828566196, −12.06547500559399, −11.50766530129320, −11.18627307630511, −10.73645519967981, −10.07371745567721, −9.902641056212045, −9.088867493226423, −8.873986470471865, −8.269977196374206, −7.621628537413393, −7.240350492297933, −6.806225611362499, −6.287246433445723, −5.622208491346583, −5.304266175657965, −4.580914806131048, −3.848634071570667, −3.464230455078506, −2.825159567488992, −2.048400915810898, −1.300777981986611, −0.7715320097915850, 0,
0.7715320097915850, 1.300777981986611, 2.048400915810898, 2.825159567488992, 3.464230455078506, 3.848634071570667, 4.580914806131048, 5.304266175657965, 5.622208491346583, 6.287246433445723, 6.806225611362499, 7.240350492297933, 7.621628537413393, 8.269977196374206, 8.873986470471865, 9.088867493226423, 9.902641056212045, 10.07371745567721, 10.73645519967981, 11.18627307630511, 11.50766530129320, 12.06547500559399, 12.50162828566196, 12.86337949030930, 13.52448538872724
