| L(s) = 1 | + 0.976·2-s − 7.04·4-s + 2.51·5-s + 23.2·7-s − 14.6·8-s + 2.45·10-s + 4.23·11-s − 37.8·13-s + 22.6·14-s + 42.0·16-s − 93.4·17-s + 109.·19-s − 17.6·20-s + 4.13·22-s + 138.·23-s − 118.·25-s − 36.9·26-s − 163.·28-s + 36.2·29-s − 313.·31-s + 158.·32-s − 91.2·34-s + 58.3·35-s + 143.·37-s + 107.·38-s − 36.8·40-s − 216.·41-s + ⋯ |
| L(s) = 1 | + 0.345·2-s − 0.880·4-s + 0.224·5-s + 1.25·7-s − 0.649·8-s + 0.0775·10-s + 0.116·11-s − 0.806·13-s + 0.433·14-s + 0.656·16-s − 1.33·17-s + 1.32·19-s − 0.197·20-s + 0.0400·22-s + 1.25·23-s − 0.949·25-s − 0.278·26-s − 1.10·28-s + 0.231·29-s − 1.81·31-s + 0.875·32-s − 0.460·34-s + 0.281·35-s + 0.639·37-s + 0.457·38-s − 0.145·40-s − 0.824·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 0.976T + 8T^{2} \) |
| 5 | \( 1 - 2.51T + 125T^{2} \) |
| 7 | \( 1 - 23.2T + 343T^{2} \) |
| 11 | \( 1 - 4.23T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 313.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 166.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 452.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 300.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 489.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 878.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 314.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.57e3T + 9.12e5T^{2} \) |
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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
−8.390962045701770982429898676629, −7.61405433266233826724517225333, −6.84491448613841211313313906098, −5.58674139681390503975664956829, −5.09673468648403302706304083318, −4.45419314619920526764447107232, −3.51278480117908813401892443161, −2.33928362593909396183357284492, −1.26879320354926792780438977673, 0,
1.26879320354926792780438977673, 2.33928362593909396183357284492, 3.51278480117908813401892443161, 4.45419314619920526764447107232, 5.09673468648403302706304083318, 5.58674139681390503975664956829, 6.84491448613841211313313906098, 7.61405433266233826724517225333, 8.390962045701770982429898676629
