| L(s) = 1 | + 1.54·2-s + 0.124·3-s + 0.377·4-s + 0.192·6-s − 4.01·7-s − 2.50·8-s − 2.98·9-s + 0.974·11-s + 0.0471·12-s + 2.01·13-s − 6.19·14-s − 4.61·16-s − 7.68·17-s − 4.60·18-s + 1.11·19-s − 0.502·21-s + 1.50·22-s + 2.43·23-s − 0.312·24-s + 3.11·26-s − 0.747·27-s − 1.51·28-s − 6.25·29-s + 31-s − 2.10·32-s + 0.121·33-s − 11.8·34-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.0721·3-s + 0.188·4-s + 0.0786·6-s − 1.51·7-s − 0.884·8-s − 0.994·9-s + 0.293·11-s + 0.0136·12-s + 0.559·13-s − 1.65·14-s − 1.15·16-s − 1.86·17-s − 1.08·18-s + 0.255·19-s − 0.109·21-s + 0.320·22-s + 0.507·23-s − 0.0638·24-s + 0.610·26-s − 0.143·27-s − 0.286·28-s − 1.16·29-s + 0.179·31-s − 0.372·32-s + 0.0211·33-s − 2.03·34-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 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 0.124T + 3T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 0.974T + 11T^{2} \) |
| 13 | \( 1 - 2.01T + 13T^{2} \) |
| 17 | \( 1 + 7.68T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 9.85T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 4.20T + 73T^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−9.640180564860557614829411849082, −9.175432515423924296775434463531, −8.322602888535907082738852264735, −6.73803002023152352594981845449, −6.30360095761504171834735993112, −5.43331805179682219880286011836, −4.26556015538163957034862653700, −3.39970284215423413799154690200, −2.59047915425360233582599871283, 0,
2.59047915425360233582599871283, 3.39970284215423413799154690200, 4.26556015538163957034862653700, 5.43331805179682219880286011836, 6.30360095761504171834735993112, 6.73803002023152352594981845449, 8.322602888535907082738852264735, 9.175432515423924296775434463531, 9.640180564860557614829411849082
