| L(s) = 1 | − 3·2-s + 5·3-s + 4-s − 15·6-s + 7-s + 21·8-s − 2·9-s − 24·11-s + 5·12-s + 31·13-s − 3·14-s − 71·16-s − 33·17-s + 6·18-s + 19·19-s + 5·21-s + 72·22-s − 27·23-s + 105·24-s − 93·26-s − 145·27-s + 28-s + 111·29-s − 94·31-s + 45·32-s − 120·33-s + 99·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.962·3-s + 1/8·4-s − 1.02·6-s + 0.0539·7-s + 0.928·8-s − 0.0740·9-s − 0.657·11-s + 0.120·12-s + 0.661·13-s − 0.0572·14-s − 1.10·16-s − 0.470·17-s + 0.0785·18-s + 0.229·19-s + 0.0519·21-s + 0.697·22-s − 0.244·23-s + 0.893·24-s − 0.701·26-s − 1.03·27-s + 0.00674·28-s + 0.710·29-s − 0.544·31-s + 0.248·32-s − 0.633·33-s + 0.499·34-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 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 31 T + p^{3} T^{2} \) |
| 17 | \( 1 + 33 T + p^{3} T^{2} \) |
| 23 | \( 1 + 27 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 94 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 510 T + p^{3} T^{2} \) |
| 43 | \( 1 - 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 75 T + p^{3} T^{2} \) |
| 59 | \( 1 - 45 T + p^{3} T^{2} \) |
| 61 | \( 1 + 28 T + p^{3} T^{2} \) |
| 67 | \( 1 + 371 T + p^{3} T^{2} \) |
| 71 | \( 1 - 384 T + p^{3} T^{2} \) |
| 73 | \( 1 - p T + p^{3} T^{2} \) |
| 79 | \( 1 + 1234 T + p^{3} T^{2} \) |
| 83 | \( 1 + 366 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1578 T + p^{3} T^{2} \) |
| 97 | \( 1 - 538 T + p^{3} T^{2} \) |
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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
−9.931393216084882490948218621125, −9.081591215168866028755752982019, −8.411605360560725508170010470462, −7.88578810573780902344788171511, −6.81584946762386867422188849783, −5.38490085292613420263912845452, −4.10183453689330288531031210781, −2.84645272220318770416902291179, −1.59894559077942505131274662004, 0,
1.59894559077942505131274662004, 2.84645272220318770416902291179, 4.10183453689330288531031210781, 5.38490085292613420263912845452, 6.81584946762386867422188849783, 7.88578810573780902344788171511, 8.411605360560725508170010470462, 9.081591215168866028755752982019, 9.931393216084882490948218621125
