| L(s) = 1 | − 1.76·2-s + 1.12·4-s − 3.31·5-s − 4.87·7-s + 1.55·8-s + 5.86·10-s + 2.38·11-s − 1.83·13-s + 8.61·14-s − 4.98·16-s + 1.94·19-s − 3.72·20-s − 4.20·22-s + 7.17·23-s + 6.02·25-s + 3.24·26-s − 5.46·28-s − 6.02·29-s − 5.65·31-s + 5.70·32-s + 16.1·35-s − 1.57·37-s − 3.43·38-s − 5.15·40-s + 10.4·41-s + 5.75·43-s + 2.66·44-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.560·4-s − 1.48·5-s − 1.84·7-s + 0.549·8-s + 1.85·10-s + 0.717·11-s − 0.509·13-s + 2.30·14-s − 1.24·16-s + 0.445·19-s − 0.831·20-s − 0.896·22-s + 1.49·23-s + 1.20·25-s + 0.636·26-s − 1.03·28-s − 1.11·29-s − 1.01·31-s + 1.00·32-s + 2.73·35-s − 0.258·37-s − 0.556·38-s − 0.815·40-s + 1.63·41-s + 0.878·43-s + 0.402·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 59 | \( 1 - 0.726T + 59T^{2} \) |
| 61 | \( 1 + 7.22T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 - 1.34T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.08T + 97T^{2} \) |
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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
−8.734851715200021769537410304009, −7.61055492954876210974200381144, −7.26279447346941200735532285660, −6.66764456723387105484673040457, −5.48181093090533781880490544870, −4.19093548321278246998733468888, −3.64323911641429004070625069644, −2.65362609531293333462633162229, −0.920339756327936327127012037049, 0,
0.920339756327936327127012037049, 2.65362609531293333462633162229, 3.64323911641429004070625069644, 4.19093548321278246998733468888, 5.48181093090533781880490544870, 6.66764456723387105484673040457, 7.26279447346941200735532285660, 7.61055492954876210974200381144, 8.734851715200021769537410304009
