| L(s) = 1 | + 1.50·2-s − 0.816·3-s + 0.252·4-s − 1.22·6-s + 7-s − 2.62·8-s − 2.33·9-s − 5.57·11-s − 0.206·12-s + 4.04·13-s + 1.50·14-s − 4.44·16-s + 6.57·17-s − 3.50·18-s + 0.0315·19-s − 0.816·21-s − 8.36·22-s + 23-s + 2.14·24-s + 6.07·26-s + 4.35·27-s + 0.252·28-s − 6.35·29-s + 1.36·31-s − 1.42·32-s + 4.55·33-s + 9.87·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.471·3-s + 0.126·4-s − 0.500·6-s + 0.377·7-s − 0.927·8-s − 0.777·9-s − 1.68·11-s − 0.0595·12-s + 1.12·13-s + 0.401·14-s − 1.11·16-s + 1.59·17-s − 0.825·18-s + 0.00724·19-s − 0.178·21-s − 1.78·22-s + 0.208·23-s + 0.437·24-s + 1.19·26-s + 0.838·27-s + 0.0477·28-s − 1.18·29-s + 0.245·31-s − 0.251·32-s + 0.792·33-s + 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956195285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956195285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 3 | \( 1 + 0.816T + 3T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 - 0.0315T + 19T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 1.93T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 1.66T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.275623869560086371432943521103, −7.81786814325249749157231008713, −6.70112200300244185867981251713, −5.81086102979218363168912148488, −5.37730118502797335992029336650, −5.01865469451524557802272139488, −3.78311825034828245425847030882, −3.23305454328870534997897446374, −2.28041378741148342546400830497, −0.68127511113872121050138131585,
0.68127511113872121050138131585, 2.28041378741148342546400830497, 3.23305454328870534997897446374, 3.78311825034828245425847030882, 5.01865469451524557802272139488, 5.37730118502797335992029336650, 5.81086102979218363168912148488, 6.70112200300244185867981251713, 7.81786814325249749157231008713, 8.275623869560086371432943521103
