| L(s) = 1 | − 2-s − 0.656·3-s + 4-s − 1.22·5-s + 0.656·6-s + 2.46·7-s − 8-s − 2.56·9-s + 1.22·10-s + 2.53·11-s − 0.656·12-s + 2.77·13-s − 2.46·14-s + 0.806·15-s + 16-s − 0.964·17-s + 2.56·18-s − 19-s − 1.22·20-s − 1.61·21-s − 2.53·22-s + 3.35·23-s + 0.656·24-s − 3.48·25-s − 2.77·26-s + 3.65·27-s + 2.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.378·3-s + 0.5·4-s − 0.549·5-s + 0.267·6-s + 0.932·7-s − 0.353·8-s − 0.856·9-s + 0.388·10-s + 0.763·11-s − 0.189·12-s + 0.770·13-s − 0.659·14-s + 0.208·15-s + 0.250·16-s − 0.233·17-s + 0.605·18-s − 0.229·19-s − 0.274·20-s − 0.353·21-s − 0.540·22-s + 0.699·23-s + 0.133·24-s − 0.697·25-s − 0.544·26-s + 0.703·27-s + 0.466·28-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 | \( 1 + 0.656T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 + 0.964T + 17T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 0.507T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 + 1.67T + 53T^{2} \) |
| 59 | \( 1 + 0.559T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 - 4.20T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 + 8.59T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 2.84T + 97T^{2} \) |
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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
−7.69741135282082785896651625650, −6.66979337011114762316171676338, −6.40142053225896605630806115365, −5.37992250521533231100987384975, −4.79340196044929899407438161801, −3.81382128238763811782007634786, −3.11349235845796247636539052395, −1.95366095900967865780194539798, −1.15496957855666958717773835475, 0,
1.15496957855666958717773835475, 1.95366095900967865780194539798, 3.11349235845796247636539052395, 3.81382128238763811782007634786, 4.79340196044929899407438161801, 5.37992250521533231100987384975, 6.40142053225896605630806115365, 6.66979337011114762316171676338, 7.69741135282082785896651625650
