L(s) = 1 | + 0.901·2-s − 0.325·3-s − 1.18·4-s + 5-s − 0.293·6-s + 7-s − 2.87·8-s − 2.89·9-s + 0.901·10-s + 1.13·11-s + 0.386·12-s + 0.315·13-s + 0.901·14-s − 0.325·15-s − 0.213·16-s + 6.64·17-s − 2.60·18-s − 5.25·19-s − 1.18·20-s − 0.325·21-s + 1.02·22-s − 8.90·23-s + 0.934·24-s + 25-s + 0.283·26-s + 1.91·27-s − 1.18·28-s + ⋯ |
L(s) = 1 | + 0.637·2-s − 0.187·3-s − 0.593·4-s + 0.447·5-s − 0.119·6-s + 0.377·7-s − 1.01·8-s − 0.964·9-s + 0.284·10-s + 0.342·11-s + 0.111·12-s + 0.0873·13-s + 0.240·14-s − 0.0839·15-s − 0.0532·16-s + 1.61·17-s − 0.614·18-s − 1.20·19-s − 0.265·20-s − 0.0709·21-s + 0.218·22-s − 1.85·23-s + 0.190·24-s + 0.200·25-s + 0.0556·26-s + 0.368·27-s − 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 0.901T + 2T^{2} \) |
| 3 | \( 1 + 0.325T + 3T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 0.315T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 + 0.210T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 0.766T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 + 5.18T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 - 0.480T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 4.73T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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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
−7.62550498266285248774811870281, −6.34791590688554103361186371393, −5.96835914295632614303607194168, −5.52331572446496916539137583577, −4.60295655155449346245603956228, −4.08010764529338172186620064877, −3.18225791901571335661884425370, −2.43083896077742932471000090820, −1.24424796123742239525731154060, 0,
1.24424796123742239525731154060, 2.43083896077742932471000090820, 3.18225791901571335661884425370, 4.08010764529338172186620064877, 4.60295655155449346245603956228, 5.52331572446496916539137583577, 5.96835914295632614303607194168, 6.34791590688554103361186371393, 7.62550498266285248774811870281