| L(s) = 1 | + (−0.762 + 1.19i)2-s + (−0.499 − 1.53i)3-s + (−0.838 − 1.81i)4-s + (1.60 − 1.55i)5-s + (2.21 + 0.576i)6-s + (−2.16 − 2.16i)7-s + (2.80 + 0.385i)8-s + (0.313 − 0.227i)9-s + (0.629 + 3.09i)10-s + (−4.88 + 0.774i)11-s + (−2.37 + 2.19i)12-s + (0.0996 + 0.137i)13-s + (4.23 − 0.929i)14-s + (−3.19 − 1.69i)15-s + (−2.59 + 3.04i)16-s + (−2.68 + 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.538 + 0.842i)2-s + (−0.288 − 0.887i)3-s + (−0.419 − 0.907i)4-s + (0.718 − 0.695i)5-s + (0.903 + 0.235i)6-s + (−0.819 − 0.819i)7-s + (0.990 + 0.136i)8-s + (0.104 − 0.0758i)9-s + (0.199 + 0.979i)10-s + (−1.47 + 0.233i)11-s + (−0.684 + 0.633i)12-s + (0.0276 + 0.0380i)13-s + (1.13 − 0.248i)14-s + (−0.824 − 0.436i)15-s + (−0.648 + 0.761i)16-s + (−0.651 + 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267776 - 0.525076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267776 - 0.525076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.762 - 1.19i)T \) |
| 5 | \( 1 + (-1.60 + 1.55i)T \) |
| good | 3 | \( 1 + (0.499 + 1.53i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (2.16 + 2.16i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.88 - 0.774i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-0.0996 - 0.137i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.68 - 1.36i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 2.94i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (5.13 - 0.812i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.41 + 2.78i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (-5.65 - 1.83i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.38 + 7.41i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 1.53i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-8.29 - 4.22i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.342 - 1.05i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.84 - 0.292i)T + (56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (0.747 - 0.118i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (8.51 + 2.76i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.536 - 1.65i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.129 - 0.819i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.479 + 1.47i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.43 + 16.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.8 + 8.61i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.03 + 7.92i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.35236493449648544688051352359, −10.17659095549719270567186255773, −9.031640138870680771099600682376, −7.943950022108240539462050094945, −7.23634566669917541037898942455, −6.27452291020191220165639896509, −5.58074221506567036851447946751, −4.26878505457529163879605576797, −1.96339716367492294161883713591, −0.45489914745643709549604233063,
2.38646841945400360717146553040, 3.13195478066068182444864109487, 4.65579983307044294686458470745, 5.67611499638397198226157781979, 6.94795257595766412800532108672, 8.205654342153289292991966210246, 9.306715871418821498009727700387, 9.939993908593691267968918188375, 10.51210001098380582322748954621, 11.23506660777312556924884801232
