| L(s) = 1 | − 2.03·2-s − 1.51·3-s + 2.15·4-s + 5-s + 3.09·6-s + 2.57·7-s − 0.307·8-s − 0.690·9-s − 2.03·10-s − 1.88·11-s − 3.26·12-s + 4.44·13-s − 5.24·14-s − 1.51·15-s − 3.67·16-s − 1.96·17-s + 1.40·18-s + 1.67·19-s + 2.15·20-s − 3.91·21-s + 3.84·22-s + 1.51·23-s + 0.467·24-s + 25-s − 9.05·26-s + 5.60·27-s + 5.54·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.877·3-s + 1.07·4-s + 0.447·5-s + 1.26·6-s + 0.973·7-s − 0.108·8-s − 0.230·9-s − 0.644·10-s − 0.568·11-s − 0.943·12-s + 1.23·13-s − 1.40·14-s − 0.392·15-s − 0.918·16-s − 0.476·17-s + 0.331·18-s + 0.383·19-s + 0.480·20-s − 0.854·21-s + 0.819·22-s + 0.316·23-s + 0.0954·24-s + 0.200·25-s − 1.77·26-s + 1.07·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 9.39T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + 0.787T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.241520049733004186765013448121, −7.47725584950072953747787869607, −6.68908510917012629473787093585, −5.94119299446439038250285160851, −5.20008067352100838750521085099, −4.51299758414817032817752545057, −3.12805402784365255079366956767, −1.90017899413792410491872737844, −1.20282522586832668508859049040, 0,
1.20282522586832668508859049040, 1.90017899413792410491872737844, 3.12805402784365255079366956767, 4.51299758414817032817752545057, 5.20008067352100838750521085099, 5.94119299446439038250285160851, 6.68908510917012629473787093585, 7.47725584950072953747787869607, 8.241520049733004186765013448121
