| L(s) = 1 | + 2.68·2-s + 5.19·4-s + 7-s + 8.56·8-s + 3.19·11-s − 6.38·13-s + 2.68·14-s + 12.5·16-s + 5.36·17-s − 2.38·19-s + 8.57·22-s − 3.19·23-s − 17.1·26-s + 5.19·28-s + 2.16·29-s + 6·31-s + 16.6·32-s + 14.3·34-s − 3·37-s − 6.39·38-s − 10.7·41-s + 9.38·43-s + 16.6·44-s − 8.57·46-s − 11.7·47-s + 49-s − 33.1·52-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.59·4-s + 0.377·7-s + 3.02·8-s + 0.964·11-s − 1.77·13-s + 0.716·14-s + 3.14·16-s + 1.30·17-s − 0.547·19-s + 1.82·22-s − 0.666·23-s − 3.35·26-s + 0.981·28-s + 0.402·29-s + 1.07·31-s + 2.93·32-s + 2.46·34-s − 0.493·37-s − 1.03·38-s − 1.67·41-s + 1.43·43-s + 2.50·44-s − 1.26·46-s − 1.71·47-s + 0.142·49-s − 4.59·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.923082111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.923082111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 4.33T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.797297514487043560388788835563, −8.323501983257592123746332283249, −7.46857881468400545133248887166, −6.76420328931820667912612815584, −5.99108612341475114309452645515, −5.07153922176558346118031639402, −4.53572064774974482894286492329, −3.59401944327287249777929363526, −2.67142635862485585553158331813, −1.63462201751847420076663340908,
1.63462201751847420076663340908, 2.67142635862485585553158331813, 3.59401944327287249777929363526, 4.53572064774974482894286492329, 5.07153922176558346118031639402, 5.99108612341475114309452645515, 6.76420328931820667912612815584, 7.46857881468400545133248887166, 8.323501983257592123746332283249, 9.797297514487043560388788835563
