| L(s) = 1 | − 1.26·2-s − 1.31·3-s − 0.401·4-s + 1.66·6-s − 1.13·7-s + 3.03·8-s − 1.26·9-s + 1.66·11-s + 0.528·12-s + 2.30·13-s + 1.43·14-s − 3.03·16-s + 0.983·17-s + 1.59·18-s + 0.370·19-s + 1.49·21-s − 2.10·22-s + 1.13·23-s − 3.99·24-s − 2.90·26-s + 5.61·27-s + 0.456·28-s − 3.49·29-s − 31-s − 2.23·32-s − 2.19·33-s − 1.24·34-s + ⋯ |
| L(s) = 1 | − 0.894·2-s − 0.760·3-s − 0.200·4-s + 0.680·6-s − 0.429·7-s + 1.07·8-s − 0.421·9-s + 0.502·11-s + 0.152·12-s + 0.638·13-s + 0.384·14-s − 0.759·16-s + 0.238·17-s + 0.376·18-s + 0.0849·19-s + 0.326·21-s − 0.449·22-s + 0.237·23-s − 0.816·24-s − 0.570·26-s + 1.08·27-s + 0.0862·28-s − 0.649·29-s − 0.179·31-s − 0.394·32-s − 0.382·33-s − 0.213·34-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 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 0.983T + 17T^{2} \) |
| 19 | \( 1 - 0.370T + 19T^{2} \) |
| 23 | \( 1 - 1.13T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 9.47T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−9.793033888160299488190670433530, −9.079507662468498703724292953858, −8.373905587843676492429492640669, −7.37187284168599697932760731330, −6.40190703697262759738551765578, −5.54648381822596807056915947014, −4.49485245595497133236860354954, −3.28496309208753781441090371834, −1.42930093500139456747876299853, 0,
1.42930093500139456747876299853, 3.28496309208753781441090371834, 4.49485245595497133236860354954, 5.54648381822596807056915947014, 6.40190703697262759738551765578, 7.37187284168599697932760731330, 8.373905587843676492429492640669, 9.079507662468498703724292953858, 9.793033888160299488190670433530
