L(s) = 1 | + 2-s − 0.219·3-s + 4-s + 2.27·5-s − 0.219·6-s − 3.32·7-s + 8-s − 2.95·9-s + 2.27·10-s + 3.87·11-s − 0.219·12-s − 1.48·13-s − 3.32·14-s − 0.498·15-s + 16-s + 0.776·17-s − 2.95·18-s + 19-s + 2.27·20-s + 0.729·21-s + 3.87·22-s − 8.89·23-s − 0.219·24-s + 0.173·25-s − 1.48·26-s + 1.30·27-s − 3.32·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.126·3-s + 0.5·4-s + 1.01·5-s − 0.0895·6-s − 1.25·7-s + 0.353·8-s − 0.983·9-s + 0.719·10-s + 1.16·11-s − 0.0633·12-s − 0.413·13-s − 0.888·14-s − 0.128·15-s + 0.250·16-s + 0.188·17-s − 0.695·18-s + 0.229·19-s + 0.508·20-s + 0.159·21-s + 0.825·22-s − 1.85·23-s − 0.0447·24-s + 0.0346·25-s − 0.292·26-s + 0.251·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 0.219T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 0.776T + 17T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 2.88T + 31T^{2} \) |
| 37 | \( 1 + 7.10T + 37T^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 - 0.180T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 + 0.157T + 67T^{2} \) |
| 71 | \( 1 - 4.50T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 0.915T + 97T^{2} \) |
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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
−7.17350935631296732793423703201, −6.47345812312853346955411952331, −6.08377230935166574395951860604, −5.63366410202951661082270028016, −4.71890122157662244501420468585, −3.80058671710974907246775933743, −3.15217956266419736329233295256, −2.40141076840785968200731565365, −1.49426203223721646308967936084, 0,
1.49426203223721646308967936084, 2.40141076840785968200731565365, 3.15217956266419736329233295256, 3.80058671710974907246775933743, 4.71890122157662244501420468585, 5.63366410202951661082270028016, 6.08377230935166574395951860604, 6.47345812312853346955411952331, 7.17350935631296732793423703201