| L(s) = 1 | + 0.818·2-s + 3-s − 1.33·4-s − 2.85·5-s + 0.818·6-s + 7-s − 2.72·8-s + 9-s − 2.33·10-s − 4.01·11-s − 1.33·12-s − 2.18·13-s + 0.818·14-s − 2.85·15-s + 0.430·16-s + 7.76·17-s + 0.818·18-s + 3.10·19-s + 3.80·20-s + 21-s − 3.28·22-s + 6.02·23-s − 2.72·24-s + 3.17·25-s − 1.78·26-s + 27-s − 1.33·28-s + ⋯ |
| L(s) = 1 | + 0.578·2-s + 0.577·3-s − 0.665·4-s − 1.27·5-s + 0.334·6-s + 0.377·7-s − 0.963·8-s + 0.333·9-s − 0.739·10-s − 1.21·11-s − 0.384·12-s − 0.604·13-s + 0.218·14-s − 0.738·15-s + 0.107·16-s + 1.88·17-s + 0.192·18-s + 0.711·19-s + 0.850·20-s + 0.218·21-s − 0.700·22-s + 1.25·23-s − 0.556·24-s + 0.634·25-s − 0.349·26-s + 0.192·27-s − 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.818T + 2T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 7.76T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 9.25T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 0.00589T + 37T^{2} \) |
| 41 | \( 1 + 9.78T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + 0.936T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 0.971T + 83T^{2} \) |
| 89 | \( 1 + 0.337T + 89T^{2} \) |
| 97 | \( 1 + 2.25T + 97T^{2} \) |
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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
−7.65991834767731550868766968881, −7.09624250544100345554131176448, −5.80337866419655383990862511635, −5.08827677570052751855557064213, −4.78576175345412200954491722536, −3.74817159370971713378792074610, −3.31714332100999138561836119627, −2.65607487628765625506069035469, −1.14667695352690214482780021524, 0,
1.14667695352690214482780021524, 2.65607487628765625506069035469, 3.31714332100999138561836119627, 3.74817159370971713378792074610, 4.78576175345412200954491722536, 5.08827677570052751855557064213, 5.80337866419655383990862511635, 7.09624250544100345554131176448, 7.65991834767731550868766968881
