| L(s) = 1 | + 1.18·2-s − 1.23·3-s − 0.602·4-s + 5-s − 1.46·6-s − 1.06·7-s − 3.07·8-s − 1.47·9-s + 1.18·10-s + 11-s + 0.744·12-s + 2.90·13-s − 1.26·14-s − 1.23·15-s − 2.43·16-s − 3.32·17-s − 1.74·18-s + 4.84·19-s − 0.602·20-s + 1.32·21-s + 1.18·22-s + 2.67·23-s + 3.80·24-s + 25-s + 3.43·26-s + 5.52·27-s + 0.644·28-s + ⋯ |
| L(s) = 1 | + 0.835·2-s − 0.713·3-s − 0.301·4-s + 0.447·5-s − 0.596·6-s − 0.403·7-s − 1.08·8-s − 0.490·9-s + 0.373·10-s + 0.301·11-s + 0.215·12-s + 0.805·13-s − 0.337·14-s − 0.319·15-s − 0.607·16-s − 0.805·17-s − 0.410·18-s + 1.11·19-s − 0.134·20-s + 0.288·21-s + 0.252·22-s + 0.557·23-s + 0.776·24-s + 0.200·25-s + 0.673·26-s + 1.06·27-s + 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 1.91T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + 3.36T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 + 0.772T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.262139502730755154207756766878, −6.93601152255546682946454633836, −6.28702907232649635904102118071, −5.89020095075932721250862766883, −5.01042833443123968981559448169, −4.55242077081451694247706789389, −3.36043222633447262194050955298, −2.89997042346824265567439800614, −1.32532833611040890107945922636, 0,
1.32532833611040890107945922636, 2.89997042346824265567439800614, 3.36043222633447262194050955298, 4.55242077081451694247706789389, 5.01042833443123968981559448169, 5.89020095075932721250862766883, 6.28702907232649635904102118071, 6.93601152255546682946454633836, 8.262139502730755154207756766878
