| L(s) = 1 | − 2·2-s + 4·4-s + 8·5-s − 22·7-s − 8·8-s − 16·10-s + 11·11-s − 54·13-s + 44·14-s + 16·16-s + 26·17-s − 38·19-s + 32·20-s − 22·22-s + 64·23-s − 61·25-s + 108·26-s − 88·28-s − 294·29-s + 36·31-s − 32·32-s − 52·34-s − 176·35-s − 390·37-s + 76·38-s − 64·40-s − 138·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.715·5-s − 1.18·7-s − 0.353·8-s − 0.505·10-s + 0.301·11-s − 1.15·13-s + 0.839·14-s + 1/4·16-s + 0.370·17-s − 0.458·19-s + 0.357·20-s − 0.213·22-s + 0.580·23-s − 0.487·25-s + 0.814·26-s − 0.593·28-s − 1.88·29-s + 0.208·31-s − 0.176·32-s − 0.262·34-s − 0.849·35-s − 1.73·37-s + 0.324·38-s − 0.252·40-s − 0.525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 64 T + p^{3} T^{2} \) |
| 29 | \( 1 + 294 T + p^{3} T^{2} \) |
| 31 | \( 1 - 36 T + p^{3} T^{2} \) |
| 37 | \( 1 + 390 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 242 T + p^{3} T^{2} \) |
| 47 | \( 1 + 132 T + p^{3} T^{2} \) |
| 53 | \( 1 + 388 T + p^{3} T^{2} \) |
| 59 | \( 1 - 732 T + p^{3} T^{2} \) |
| 61 | \( 1 - 430 T + p^{3} T^{2} \) |
| 67 | \( 1 - 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 420 T + p^{3} T^{2} \) |
| 73 | \( 1 + 594 T + p^{3} T^{2} \) |
| 79 | \( 1 - 506 T + p^{3} T^{2} \) |
| 83 | \( 1 + 380 T + p^{3} T^{2} \) |
| 89 | \( 1 - 256 T + p^{3} T^{2} \) |
| 97 | \( 1 - 418 T + p^{3} T^{2} \) |
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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
−11.45082146624960197892788494411, −10.10873589870169347970236787717, −9.723199550212738667314221069381, −8.767206004501656409990464211957, −7.34242372685585983404897740582, −6.49471421146267935633847462503, −5.35090919676055990888236073442, −3.42046759112430922416788234432, −1.99647259534230202894205223117, 0,
1.99647259534230202894205223117, 3.42046759112430922416788234432, 5.35090919676055990888236073442, 6.49471421146267935633847462503, 7.34242372685585983404897740582, 8.767206004501656409990464211957, 9.723199550212738667314221069381, 10.10873589870169347970236787717, 11.45082146624960197892788494411
