| L(s) = 1 | − 2.58·2-s − 2.24·3-s + 4.67·4-s + 5.78·6-s + 2.11·7-s − 6.91·8-s + 2.01·9-s − 4.70·11-s − 10.4·12-s + 3.53·13-s − 5.45·14-s + 8.52·16-s − 6.45·17-s − 5.21·18-s + 0.766·19-s − 4.73·21-s + 12.1·22-s + 5.38·23-s + 15.4·24-s − 9.13·26-s + 2.19·27-s + 9.87·28-s + 8.03·29-s − 31-s − 8.18·32-s + 10.5·33-s + 16.6·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.29·3-s + 2.33·4-s + 2.36·6-s + 0.798·7-s − 2.44·8-s + 0.672·9-s − 1.41·11-s − 3.02·12-s + 0.980·13-s − 1.45·14-s + 2.13·16-s − 1.56·17-s − 1.22·18-s + 0.175·19-s − 1.03·21-s + 2.59·22-s + 1.12·23-s + 3.16·24-s − 1.79·26-s + 0.422·27-s + 1.86·28-s + 1.49·29-s − 0.179·31-s − 1.44·32-s + 1.83·33-s + 2.86·34-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 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 - 2.11T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 - 0.766T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 + 3.15T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 0.211T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−10.21494755728651771647988526252, −8.707056701313608823166230026647, −8.545540838397914049702609186184, −7.32182144565475562505521391674, −6.67747042635238370976883676253, −5.67651502970286900667213634735, −4.75235670584499975437903386259, −2.68061690539042770811739280582, −1.33052268647320693995357161692, 0,
1.33052268647320693995357161692, 2.68061690539042770811739280582, 4.75235670584499975437903386259, 5.67651502970286900667213634735, 6.67747042635238370976883676253, 7.32182144565475562505521391674, 8.545540838397914049702609186184, 8.707056701313608823166230026647, 10.21494755728651771647988526252
