L(s) = 1 | + 2-s − 0.909·3-s + 4-s + 3.81·5-s − 0.909·6-s + 4.56·7-s + 8-s − 2.17·9-s + 3.81·10-s − 3.92·11-s − 0.909·12-s + 5.78·13-s + 4.56·14-s − 3.46·15-s + 16-s − 2.17·17-s − 2.17·18-s − 5.55·19-s + 3.81·20-s − 4.15·21-s − 3.92·22-s − 7.56·23-s − 0.909·24-s + 9.54·25-s + 5.78·26-s + 4.70·27-s + 4.56·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.525·3-s + 0.5·4-s + 1.70·5-s − 0.371·6-s + 1.72·7-s + 0.353·8-s − 0.724·9-s + 1.20·10-s − 1.18·11-s − 0.262·12-s + 1.60·13-s + 1.21·14-s − 0.895·15-s + 0.250·16-s − 0.528·17-s − 0.512·18-s − 1.27·19-s + 0.852·20-s − 0.905·21-s − 0.837·22-s − 1.57·23-s − 0.185·24-s + 1.90·25-s + 1.13·26-s + 0.905·27-s + 0.862·28-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
\(L(1)\) |
\(\approx\) |
\(4.093197828\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.093197828\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.909T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 5.62T + 31T^{2} \) |
| 37 | \( 1 - 0.397T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 5.40T + 43T^{2} \) |
| 47 | \( 1 + 1.01T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.51T + 67T^{2} \) |
| 71 | \( 1 + 5.97T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 - 5.78T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−8.392466293129284293202920445604, −7.84708945107732803711904039847, −6.42622068042679267281502995345, −6.09666838104988963527371097372, −5.53605052296258746702270310336, −4.84741445896309585650018219151, −4.17722674541979563428627316429, −2.60370299989089082217431806507, −2.16446680892596964874738931298, −1.15783037636444500962678564745,
1.15783037636444500962678564745, 2.16446680892596964874738931298, 2.60370299989089082217431806507, 4.17722674541979563428627316429, 4.84741445896309585650018219151, 5.53605052296258746702270310336, 6.09666838104988963527371097372, 6.42622068042679267281502995345, 7.84708945107732803711904039847, 8.392466293129284293202920445604