| L(s) = 1 | − 1.56·2-s + 0.438·4-s − 5-s − 7-s + 2.43·8-s + 1.56·10-s − 5.56·13-s + 1.56·14-s − 4.68·16-s + 4.12·17-s − 6·19-s − 0.438·20-s + 4·23-s − 4·25-s + 8.68·26-s − 0.438·28-s + 6.68·29-s + 8.24·31-s + 2.43·32-s − 6.43·34-s + 35-s − 2.68·37-s + 9.36·38-s − 2.43·40-s − 7.56·41-s + 5.68·43-s − 6.24·46-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.219·4-s − 0.447·5-s − 0.377·7-s + 0.862·8-s + 0.493·10-s − 1.54·13-s + 0.417·14-s − 1.17·16-s + 0.999·17-s − 1.37·19-s − 0.0980·20-s + 0.834·23-s − 0.800·25-s + 1.70·26-s − 0.0828·28-s + 1.24·29-s + 1.48·31-s + 0.431·32-s − 1.10·34-s + 0.169·35-s − 0.441·37-s + 1.51·38-s − 0.385·40-s − 1.18·41-s + 0.866·43-s − 0.920·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 + 8.80T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−7.72894694344073001644655368905, −6.98295544171957194561186503226, −6.50604798455828299335612892835, −5.32480488604221930815421168976, −4.69114749543458626542530833447, −3.98636373171383375777664275612, −2.92251207476484009329086240965, −2.11845878038613093023614764283, −0.926747298591365241592579972238, 0,
0.926747298591365241592579972238, 2.11845878038613093023614764283, 2.92251207476484009329086240965, 3.98636373171383375777664275612, 4.69114749543458626542530833447, 5.32480488604221930815421168976, 6.50604798455828299335612892835, 6.98295544171957194561186503226, 7.72894694344073001644655368905
