| L(s) = 1 | + 3·2-s + 5·3-s + 4-s + 15·6-s − 11·7-s − 21·8-s − 2·9-s − 54·11-s + 5·12-s − 11·13-s − 33·14-s − 71·16-s + 93·17-s − 6·18-s + 19·19-s − 55·21-s − 162·22-s − 183·23-s − 105·24-s − 33·26-s − 145·27-s − 11·28-s − 249·29-s + 56·31-s − 45·32-s − 270·33-s + 279·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.962·3-s + 1/8·4-s + 1.02·6-s − 0.593·7-s − 0.928·8-s − 0.0740·9-s − 1.48·11-s + 0.120·12-s − 0.234·13-s − 0.629·14-s − 1.10·16-s + 1.32·17-s − 0.0785·18-s + 0.229·19-s − 0.571·21-s − 1.56·22-s − 1.65·23-s − 0.893·24-s − 0.248·26-s − 1.03·27-s − 0.0742·28-s − 1.59·29-s + 0.324·31-s − 0.248·32-s − 1.42·33-s + 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 + 11 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 - 250 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 196 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 435 T + p^{3} T^{2} \) |
| 59 | \( 1 - 195 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 961 T + p^{3} T^{2} \) |
| 71 | \( 1 + 246 T + p^{3} T^{2} \) |
| 73 | \( 1 + 353 T + p^{3} T^{2} \) |
| 79 | \( 1 + 34 T + p^{3} T^{2} \) |
| 83 | \( 1 + 234 T + p^{3} T^{2} \) |
| 89 | \( 1 + 168 T + p^{3} T^{2} \) |
| 97 | \( 1 + 758 T + p^{3} T^{2} \) |
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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
−9.898773389895146872453014347188, −9.399482049321452481766541667549, −8.143879073534014866357012053441, −7.60434743653416605587020334390, −5.98804543193815382462721208157, −5.42131313607305216347656359995, −4.08483492710352013369568714694, −3.18278264450035621824758998987, −2.41903287033065926606537622001, 0,
2.41903287033065926606537622001, 3.18278264450035621824758998987, 4.08483492710352013369568714694, 5.42131313607305216347656359995, 5.98804543193815382462721208157, 7.60434743653416605587020334390, 8.143879073534014866357012053441, 9.399482049321452481766541667549, 9.898773389895146872453014347188
