| L(s) = 1 | + 1.41·2-s − 1.93·3-s − 2.73·6-s + 2.44·7-s − 2.82·8-s + 0.732·9-s − 1.26·11-s + 2.44·13-s + 3.46·14-s − 4.00·16-s − 4.38·17-s + 1.03·18-s − 7.19·19-s − 4.73·21-s − 1.79·22-s − 1.41·23-s + 5.46·24-s + 3.46·26-s + 4.38·27-s − 8.19·29-s + 31-s + 2.44·33-s − 6.19·34-s − 3.34·37-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 1.11·3-s − 1.11·6-s + 0.925·7-s − 0.999·8-s + 0.244·9-s − 0.382·11-s + 0.679·13-s + 0.925·14-s − 1.00·16-s − 1.06·17-s + 0.244·18-s − 1.65·19-s − 1.03·21-s − 0.382·22-s − 0.294·23-s + 1.11·24-s + 0.679·26-s + 0.843·27-s − 1.52·29-s + 0.179·31-s + 0.426·33-s − 1.06·34-s − 0.550·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 0.240T + 43T^{2} \) |
| 47 | \( 1 + 5.93T + 47T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 - 4.00T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 - 3.10T + 97T^{2} \) |
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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
−10.24037197173313189562257872740, −8.832085325642567596994844310632, −8.311499351887326817002586293495, −6.81205614545091664120647208536, −6.09928345293550757267181839711, −5.26718219411403071859663382111, −4.64057857094021062553182422231, −3.70594861730306321204256788191, −2.07870229890795109176940393394, 0,
2.07870229890795109176940393394, 3.70594861730306321204256788191, 4.64057857094021062553182422231, 5.26718219411403071859663382111, 6.09928345293550757267181839711, 6.81205614545091664120647208536, 8.311499351887326817002586293495, 8.832085325642567596994844310632, 10.24037197173313189562257872740
