| L(s) = 1 | + 2-s + 4-s + 3.15·5-s − 4.69·7-s + 8-s + 3.15·10-s − 0.137·11-s − 4.69·14-s + 16-s + 5.60·17-s − 4.98·19-s + 3.15·20-s − 0.137·22-s + 6.09·23-s + 4.97·25-s − 4.69·28-s + 0.850·29-s + 6.23·31-s + 32-s + 5.60·34-s − 14.8·35-s + 11.7·37-s − 4.98·38-s + 3.15·40-s + 4.27·41-s − 2.09·43-s − 0.137·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.41·5-s − 1.77·7-s + 0.353·8-s + 0.998·10-s − 0.0413·11-s − 1.25·14-s + 0.250·16-s + 1.35·17-s − 1.14·19-s + 0.706·20-s − 0.0292·22-s + 1.27·23-s + 0.995·25-s − 0.886·28-s + 0.157·29-s + 1.11·31-s + 0.176·32-s + 0.961·34-s − 2.50·35-s + 1.92·37-s − 0.809·38-s + 0.499·40-s + 0.667·41-s − 0.319·43-s − 0.0206·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.321874788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.321874788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.15T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 + 0.137T + 11T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 - 0.850T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 - 0.0978T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−8.943141890636860295357320999128, −7.84307646068883609982514363144, −6.77692884373167467615924820139, −6.33350151392703872128613239494, −5.81740366709497716364741588753, −5.00743096521024373364367139505, −3.90933235807965602446349059993, −2.95609531387861872304480538866, −2.45100061847091525661344848353, −1.02761441849840946925621366088,
1.02761441849840946925621366088, 2.45100061847091525661344848353, 2.95609531387861872304480538866, 3.90933235807965602446349059993, 5.00743096521024373364367139505, 5.81740366709497716364741588753, 6.33350151392703872128613239494, 6.77692884373167467615924820139, 7.84307646068883609982514363144, 8.943141890636860295357320999128
