| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 11-s − 12-s − 2·13-s + 14-s + 16-s + 2·17-s + 18-s − 8·19-s − 21-s − 22-s + 8·23-s − 24-s − 2·26-s − 27-s + 28-s − 6·29-s + 32-s + 33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.218·21-s − 0.213·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.70529031545045, −16.35892417530163, −15.18721879201628, −14.99237004621346, −14.76880750346867, −13.72387836866048, −13.20415491807420, −12.74386178942611, −12.24349476353412, −11.51821723710649, −11.02769520345112, −10.58985579142006, −9.885482206233393, −9.162190172119656, −8.347365271535039, −7.752435611395443, −6.958454464126762, −6.567197018733562, −5.680659249204317, −5.192791082939401, −4.563713317957722, −3.931364627751859, −2.980315804851857, −2.202111510782049, −1.302256353297105, 0,
1.302256353297105, 2.202111510782049, 2.980315804851857, 3.931364627751859, 4.563713317957722, 5.192791082939401, 5.680659249204317, 6.567197018733562, 6.958454464126762, 7.752435611395443, 8.347365271535039, 9.162190172119656, 9.885482206233393, 10.58985579142006, 11.02769520345112, 11.51821723710649, 12.24349476353412, 12.74386178942611, 13.20415491807420, 13.72387836866048, 14.76880750346867, 14.99237004621346, 15.18721879201628, 16.35892417530163, 16.70529031545045
