| L(s) = 1 | − 1.30·2-s − 2.43·3-s − 0.285·4-s + 3.19·6-s − 7-s + 2.99·8-s + 2.95·9-s + 0.628·11-s + 0.697·12-s + 3.82·13-s + 1.30·14-s − 3.34·16-s − 5.22·17-s − 3.86·18-s + 6.31·19-s + 2.43·21-s − 0.823·22-s − 23-s − 7.30·24-s − 5.00·26-s + 0.115·27-s + 0.285·28-s + 3.73·29-s + 9.35·31-s − 1.60·32-s − 1.53·33-s + 6.83·34-s + ⋯ |
| L(s) = 1 | − 0.925·2-s − 1.40·3-s − 0.142·4-s + 1.30·6-s − 0.377·7-s + 1.05·8-s + 0.984·9-s + 0.189·11-s + 0.201·12-s + 1.05·13-s + 0.349·14-s − 0.836·16-s − 1.26·17-s − 0.911·18-s + 1.44·19-s + 0.532·21-s − 0.175·22-s − 0.208·23-s − 1.49·24-s − 0.981·26-s + 0.0221·27-s + 0.0540·28-s + 0.694·29-s + 1.68·31-s − 0.283·32-s − 0.267·33-s + 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5705645500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5705645500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 11 | \( 1 - 0.628T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 9.35T + 31T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 8.98T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 6.33T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 0.664T + 83T^{2} \) |
| 89 | \( 1 + 9.82T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.390709984414795882133938936314, −7.88113975657235259227912478368, −6.81014591837476598422799370062, −6.36751367109590267810662675159, −5.61240914231049407145807517091, −4.68144755213646776072703819815, −4.16445250081745056773194141495, −2.83420490768244358119116242390, −1.31304291008016897001430381262, −0.62189819508672892075714741404,
0.62189819508672892075714741404, 1.31304291008016897001430381262, 2.83420490768244358119116242390, 4.16445250081745056773194141495, 4.68144755213646776072703819815, 5.61240914231049407145807517091, 6.36751367109590267810662675159, 6.81014591837476598422799370062, 7.88113975657235259227912478368, 8.390709984414795882133938936314
