| L(s) = 1 | + (0.913 + 0.406i)2-s + (−2.15 + 1.56i)3-s + (0.669 + 0.743i)4-s + (−2.96 + 0.630i)5-s + (−2.60 + 0.553i)6-s + (−0.177 + 1.68i)7-s + (0.309 + 0.951i)8-s + (1.26 − 3.89i)9-s + (−2.96 − 0.630i)10-s + 5.25·11-s + (−2.60 − 0.553i)12-s + (0.998 + 1.73i)13-s + (−0.847 + 1.46i)14-s + (5.40 − 6.00i)15-s + (−0.104 + 0.994i)16-s + (−2.49 − 2.77i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (−1.24 + 0.903i)3-s + (0.334 + 0.371i)4-s + (−1.32 + 0.281i)5-s + (−1.06 + 0.226i)6-s + (−0.0669 + 0.636i)7-s + (0.109 + 0.336i)8-s + (0.421 − 1.29i)9-s + (−0.937 − 0.199i)10-s + 1.58·11-s + (−0.752 − 0.159i)12-s + (0.277 + 0.479i)13-s + (−0.226 + 0.392i)14-s + (1.39 − 1.54i)15-s + (−0.0261 + 0.248i)16-s + (−0.605 − 0.672i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.379886 + 0.750348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379886 + 0.750348i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 61 | \( 1 + (-0.465 + 7.79i)T \) |
| good | 3 | \( 1 + (2.15 - 1.56i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (2.96 - 0.630i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (0.177 - 1.68i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 + (-0.998 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.49 + 2.77i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.831 - 7.90i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.499 + 1.53i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.47 + 3.32i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (4.02 + 2.92i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.56 + 1.13i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (8.30 - 9.22i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.57 + 4.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.78 - 5.49i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.67 - 3.41i)T + (39.4 + 43.8i)T^{2} \) |
| 67 | \( 1 + (-11.6 + 2.47i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (11.0 + 2.35i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-9.73 - 2.06i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (5.31 - 5.90i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (4.31 + 1.92i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.591 + 0.429i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.94 + 2.64i)T + (64.9 - 72.0i)T^{2} \) |
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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
−14.06066403343687436500444238845, −12.18752128199333968519279126999, −11.81019751739633736080926326205, −11.19873313781965811744945504119, −9.811726339380771249136077974633, −8.403980469475845365029483311402, −6.83040783656651942637069383325, −5.92166730161411920665933276347, −4.48869472609077401441721329631, −3.74122642909785943988546318552,
0.941017614271426054992161377660, 3.79416411605759084592177191891, 4.95145011999063205880371057384, 6.57290092414655073941327743699, 7.08483251297576612974560323530, 8.627028858650302506191419618042, 10.55276333683422269684217604529, 11.60025838205968064574764125852, 11.80510261549417568065853852099, 12.88696313108176847255600150842
