| L(s) = 1 | − 2.52·2-s − 1.63·4-s + 2.94·5-s − 33.1·7-s + 24.3·8-s − 7.44·10-s − 63.4·11-s + 1.98·13-s + 83.6·14-s − 48.2·16-s + 58.9·17-s + 110.·19-s − 4.81·20-s + 160.·22-s − 45.7·23-s − 116.·25-s − 5.01·26-s + 54.1·28-s − 69.0·29-s − 142.·31-s − 72.6·32-s − 148.·34-s − 97.7·35-s + 110.·37-s − 277.·38-s + 71.6·40-s + 249.·41-s + ⋯ |
| L(s) = 1 | − 0.892·2-s − 0.204·4-s + 0.263·5-s − 1.79·7-s + 1.07·8-s − 0.235·10-s − 1.73·11-s + 0.0424·13-s + 1.59·14-s − 0.754·16-s + 0.840·17-s + 1.32·19-s − 0.0538·20-s + 1.55·22-s − 0.414·23-s − 0.930·25-s − 0.0378·26-s + 0.365·28-s − 0.441·29-s − 0.825·31-s − 0.401·32-s − 0.750·34-s − 0.472·35-s + 0.492·37-s − 1.18·38-s + 0.283·40-s + 0.948·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 + 2.52T + 8T^{2} \) |
| 5 | \( 1 - 2.94T + 125T^{2} \) |
| 7 | \( 1 + 33.1T + 343T^{2} \) |
| 11 | \( 1 + 63.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.98T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 243.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 570.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 461.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 741.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 34.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 381.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 48.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 381.T + 9.12e5T^{2} \) |
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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
−8.382445285325751681651536050015, −7.53872565115454272166997438049, −7.17306454137541517057178317458, −5.72076230818364412776503131252, −5.54574845497887011147266588176, −4.11507017894554500614917858246, −3.20383986061971750724387197750, −2.31535540326776928842805611578, −0.827950147621864179190104000803, 0,
0.827950147621864179190104000803, 2.31535540326776928842805611578, 3.20383986061971750724387197750, 4.11507017894554500614917858246, 5.54574845497887011147266588176, 5.72076230818364412776503131252, 7.17306454137541517057178317458, 7.53872565115454272166997438049, 8.382445285325751681651536050015
