L(s) = 1 | + 2-s + 2.23·3-s + 4-s − 0.610·5-s + 2.23·6-s + 2.62·7-s + 8-s + 1.99·9-s − 0.610·10-s − 4.09·11-s + 2.23·12-s − 4.70·13-s + 2.62·14-s − 1.36·15-s + 16-s − 7.22·17-s + 1.99·18-s − 19-s − 0.610·20-s + 5.86·21-s − 4.09·22-s − 2.37·23-s + 2.23·24-s − 4.62·25-s − 4.70·26-s − 2.25·27-s + 2.62·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.273·5-s + 0.912·6-s + 0.991·7-s + 0.353·8-s + 0.664·9-s − 0.193·10-s − 1.23·11-s + 0.645·12-s − 1.30·13-s + 0.700·14-s − 0.352·15-s + 0.250·16-s − 1.75·17-s + 0.469·18-s − 0.229·19-s − 0.136·20-s + 1.27·21-s − 0.872·22-s − 0.495·23-s + 0.456·24-s − 0.925·25-s − 0.923·26-s − 0.433·27-s + 0.495·28-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 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 0.610T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 - 0.375T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 - 7.40T + 59T^{2} \) |
| 61 | \( 1 + 3.08T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 + 2.55T + 83T^{2} \) |
| 89 | \( 1 + 8.75T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−7.63415434344886162661686895617, −7.00998236040872398213817292958, −6.02841455113431402309644049904, −5.11399259963106351839220935704, −4.58825270419838827575054149161, −3.98189800998292411383243809856, −2.97748851919175333552880613689, −2.28950381699062889910988195755, −1.94941577763168383186011337494, 0,
1.94941577763168383186011337494, 2.28950381699062889910988195755, 2.97748851919175333552880613689, 3.98189800998292411383243809856, 4.58825270419838827575054149161, 5.11399259963106351839220935704, 6.02841455113431402309644049904, 7.00998236040872398213817292958, 7.63415434344886162661686895617