| L(s) = 1 | + 2.25·2-s + 3-s + 3.06·4-s − 2.87·5-s + 2.25·6-s − 3.82·7-s + 2.40·8-s + 9-s − 6.46·10-s − 2.53·11-s + 3.06·12-s + 1.24·13-s − 8.62·14-s − 2.87·15-s − 0.723·16-s + 17-s + 2.25·18-s + 6.75·19-s − 8.81·20-s − 3.82·21-s − 5.70·22-s − 6.24·23-s + 2.40·24-s + 3.24·25-s + 2.80·26-s + 27-s − 11.7·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.53·4-s − 1.28·5-s + 0.919·6-s − 1.44·7-s + 0.850·8-s + 0.333·9-s − 2.04·10-s − 0.763·11-s + 0.885·12-s + 0.345·13-s − 2.30·14-s − 0.741·15-s − 0.180·16-s + 0.242·17-s + 0.530·18-s + 1.54·19-s − 1.97·20-s − 0.835·21-s − 1.21·22-s − 1.30·23-s + 0.490·24-s + 0.649·25-s + 0.550·26-s + 0.192·27-s − 2.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.722641862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.722641862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 - 3.49T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 5.81T + 59T^{2} \) |
| 61 | \( 1 - 8.23T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 9.33T + 73T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.72T + 97T^{2} \) |
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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
−7.47774461507553017198945825804, −7.20142991271796298320578435161, −6.25149747335734784335585893933, −5.74672894135870599881968981057, −4.83556993962025825452972551151, −4.07388790131695783993769734462, −3.59949506335089389541363168477, −3.00585597338922843500344882524, −2.46402585794838320599058281824, −0.69666200160740587976896254440,
0.69666200160740587976896254440, 2.46402585794838320599058281824, 3.00585597338922843500344882524, 3.59949506335089389541363168477, 4.07388790131695783993769734462, 4.83556993962025825452972551151, 5.74672894135870599881968981057, 6.25149747335734784335585893933, 7.20142991271796298320578435161, 7.47774461507553017198945825804
