| L(s) = 1 | − 2-s + 1.89·3-s + 4-s + 0.229·5-s − 1.89·6-s − 7-s − 8-s + 0.588·9-s − 0.229·10-s − 2.21·11-s + 1.89·12-s − 0.819·13-s + 14-s + 0.433·15-s + 16-s + 1.55·17-s − 0.588·18-s + 3.81·19-s + 0.229·20-s − 1.89·21-s + 2.21·22-s − 1.78·23-s − 1.89·24-s − 4.94·25-s + 0.819·26-s − 4.56·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.09·3-s + 0.5·4-s + 0.102·5-s − 0.773·6-s − 0.377·7-s − 0.353·8-s + 0.196·9-s − 0.0724·10-s − 0.667·11-s + 0.546·12-s − 0.227·13-s + 0.267·14-s + 0.112·15-s + 0.250·16-s + 0.376·17-s − 0.138·18-s + 0.876·19-s + 0.0512·20-s − 0.413·21-s + 0.472·22-s − 0.372·23-s − 0.386·24-s − 0.989·25-s + 0.160·26-s − 0.879·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 - 0.229T + 5T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 + 0.819T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 + 2.64T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 + 6.16T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 6.69T + 97T^{2} \) |
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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
−7.74192713083506531291022245003, −7.54383873490315667667889190721, −6.31483631136484280330691346108, −5.81741631610740507469595516995, −4.78617723907243869555700612477, −3.74397089865772893890195682473, −2.96031053008358117513095358897, −2.45025662468297978834510361422, −1.41228318490424093998395733228, 0,
1.41228318490424093998395733228, 2.45025662468297978834510361422, 2.96031053008358117513095358897, 3.74397089865772893890195682473, 4.78617723907243869555700612477, 5.81741631610740507469595516995, 6.31483631136484280330691346108, 7.54383873490315667667889190721, 7.74192713083506531291022245003
