| L(s) = 1 | + 0.888·2-s − 1.66·3-s − 1.21·4-s − 1.48·6-s − 1.49·7-s − 2.85·8-s − 0.212·9-s − 4.65·11-s + 2.02·12-s + 6.18·13-s − 1.32·14-s − 0.114·16-s + 2.17·17-s − 0.188·18-s − 6.25·19-s + 2.48·21-s − 4.14·22-s + 7.34·23-s + 4.76·24-s + 5.49·26-s + 5.36·27-s + 1.80·28-s + 9.57·29-s + 5.04·31-s + 5.60·32-s + 7.78·33-s + 1.93·34-s + ⋯ |
| L(s) = 1 | + 0.628·2-s − 0.963·3-s − 0.605·4-s − 0.605·6-s − 0.563·7-s − 1.00·8-s − 0.0707·9-s − 1.40·11-s + 0.583·12-s + 1.71·13-s − 0.353·14-s − 0.0285·16-s + 0.527·17-s − 0.0444·18-s − 1.43·19-s + 0.543·21-s − 0.882·22-s + 1.53·23-s + 0.972·24-s + 1.07·26-s + 1.03·27-s + 0.340·28-s + 1.77·29-s + 0.906·31-s + 0.990·32-s + 1.35·33-s + 0.331·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 0.888T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 4.65T + 11T^{2} \) |
| 13 | \( 1 - 6.18T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 - 9.57T + 29T^{2} \) |
| 31 | \( 1 - 5.04T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 41 | \( 1 - 2.18T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 3.02T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−8.107475801174366541516640264067, −6.76631451477647895754894306388, −6.21559601712582150358103711184, −5.72899979706449684083353469183, −4.93811876360822028734067035035, −4.43747456478080774070312893751, −3.28481236341340414103504281276, −2.82000160811516143537064099664, −1.03804403141910549392797149566, 0,
1.03804403141910549392797149566, 2.82000160811516143537064099664, 3.28481236341340414103504281276, 4.43747456478080774070312893751, 4.93811876360822028734067035035, 5.72899979706449684083353469183, 6.21559601712582150358103711184, 6.76631451477647895754894306388, 8.107475801174366541516640264067
