| L(s) = 1 | + 2.57·3-s + 0.163·5-s + 27.8·7-s − 20.3·9-s − 24.5·11-s − 48.6·13-s + 0.420·15-s − 40.4·17-s − 6.50·19-s + 71.5·21-s + 85.1·23-s − 124.·25-s − 121.·27-s − 269.·29-s − 139.·31-s − 63.1·33-s + 4.54·35-s − 267.·37-s − 125.·39-s + 265.·41-s + 501.·43-s − 3.32·45-s + 143.·47-s + 430.·49-s − 104.·51-s − 10.0·53-s − 4.00·55-s + ⋯ |
| L(s) = 1 | + 0.495·3-s + 0.0146·5-s + 1.50·7-s − 0.754·9-s − 0.672·11-s − 1.03·13-s + 0.00723·15-s − 0.577·17-s − 0.0784·19-s + 0.743·21-s + 0.772·23-s − 0.999·25-s − 0.869·27-s − 1.72·29-s − 0.807·31-s − 0.332·33-s + 0.0219·35-s − 1.18·37-s − 0.513·39-s + 1.01·41-s + 1.77·43-s − 0.0110·45-s + 0.444·47-s + 1.25·49-s − 0.285·51-s − 0.0259·53-s − 0.00982·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 89 | \( 1 + 89T \) |
| good | 3 | \( 1 - 2.57T + 27T^{2} \) |
| 5 | \( 1 - 0.163T + 125T^{2} \) |
| 7 | \( 1 - 27.8T + 343T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.50T + 6.85e3T^{2} \) |
| 23 | \( 1 - 85.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 52.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 463.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 56.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 834.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 180.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 292.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 177.T + 5.71e5T^{2} \) |
| 97 | \( 1 - 837.T + 9.12e5T^{2} \) |
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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
−9.346607484870311424002127123379, −8.787818598915212164920657719282, −7.69815235025392319273861713069, −7.45280375033827964908914315645, −5.75721155488625001308535549377, −5.09669817414743533771269253308, −4.04525985957906341188294025911, −2.64359757601414269169048798386, −1.83260374587163647096602317213, 0,
1.83260374587163647096602317213, 2.64359757601414269169048798386, 4.04525985957906341188294025911, 5.09669817414743533771269253308, 5.75721155488625001308535549377, 7.45280375033827964908914315645, 7.69815235025392319273861713069, 8.787818598915212164920657719282, 9.346607484870311424002127123379
