| L(s) = 1 | − 2.66·2-s − 0.759·3-s + 5.10·4-s + 2.02·6-s + 2.04·7-s − 8.26·8-s − 2.42·9-s + 1.34·11-s − 3.87·12-s − 1.31·13-s − 5.44·14-s + 11.8·16-s − 4.08·17-s + 6.45·18-s − 4.88·19-s − 1.55·21-s − 3.59·22-s + 2.73·23-s + 6.27·24-s + 3.51·26-s + 4.11·27-s + 10.4·28-s − 4.61·29-s + 7.15·31-s − 14.9·32-s − 1.02·33-s + 10.8·34-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.438·3-s + 2.55·4-s + 0.825·6-s + 0.771·7-s − 2.92·8-s − 0.807·9-s + 0.406·11-s − 1.11·12-s − 0.366·13-s − 1.45·14-s + 2.95·16-s − 0.991·17-s + 1.52·18-s − 1.12·19-s − 0.338·21-s − 0.766·22-s + 0.570·23-s + 1.28·24-s + 0.689·26-s + 0.792·27-s + 1.96·28-s − 0.857·29-s + 1.28·31-s − 2.64·32-s − 0.178·33-s + 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 0.759T + 3T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 - 0.621T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.647T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.95T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−10.15349132578451826004961378131, −9.149153815130528299180056151394, −8.530176603709353931658816635295, −7.84733052529341141805298046130, −6.74976349858057710195217210491, −6.11385757440052154128364671853, −4.71233249938837677326871149431, −2.78750471905394178744992700911, −1.62168914287578897386004849686, 0,
1.62168914287578897386004849686, 2.78750471905394178744992700911, 4.71233249938837677326871149431, 6.11385757440052154128364671853, 6.74976349858057710195217210491, 7.84733052529341141805298046130, 8.530176603709353931658816635295, 9.149153815130528299180056151394, 10.15349132578451826004961378131
