L(s) = 1 | + 2-s + 2.10·3-s + 4-s − 2.78·5-s + 2.10·6-s + 2.81·7-s + 8-s + 1.43·9-s − 2.78·10-s − 5.73·11-s + 2.10·12-s − 5.97·13-s + 2.81·14-s − 5.86·15-s + 16-s − 0.579·17-s + 1.43·18-s − 1.97·19-s − 2.78·20-s + 5.92·21-s − 5.73·22-s + 2.52·23-s + 2.10·24-s + 2.73·25-s − 5.97·26-s − 3.28·27-s + 2.81·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.21·3-s + 0.5·4-s − 1.24·5-s + 0.860·6-s + 1.06·7-s + 0.353·8-s + 0.479·9-s − 0.879·10-s − 1.72·11-s + 0.608·12-s − 1.65·13-s + 0.751·14-s − 1.51·15-s + 0.250·16-s − 0.140·17-s + 0.339·18-s − 0.454·19-s − 0.621·20-s + 1.29·21-s − 1.22·22-s + 0.527·23-s + 0.430·24-s + 0.547·25-s − 1.17·26-s − 0.632·27-s + 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 + 0.579T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 0.807T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 + 0.432T + 43T^{2} \) |
| 47 | \( 1 - 0.866T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 0.972T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 + 4.66T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 + 9.57T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.15T + 89T^{2} \) |
| 97 | \( 1 - 0.840T + 97T^{2} \) |
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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
−8.064171761005832984215369420514, −7.52456332329420296714551074190, −6.97460872996592486933580202765, −5.51139803286619211794986329231, −4.79271026879412734200570067203, −4.38973034729171501600050520019, −3.24752533286481575999912471635, −2.71321033975993273761428484453, −1.93305671099930912052931283608, 0,
1.93305671099930912052931283608, 2.71321033975993273761428484453, 3.24752533286481575999912471635, 4.38973034729171501600050520019, 4.79271026879412734200570067203, 5.51139803286619211794986329231, 6.97460872996592486933580202765, 7.52456332329420296714551074190, 8.064171761005832984215369420514