| L(s) = 1 | + (0.523 − 1.31i)2-s + (2.69 − 0.304i)3-s + (−1.45 − 1.37i)4-s + (−1.20 + 2.49i)5-s + (1.01 − 3.70i)6-s + (−0.625 − 0.498i)7-s + (−2.56 + 1.18i)8-s + (4.27 − 0.974i)9-s + (2.64 + 2.88i)10-s + (−4.00 + 2.51i)11-s + (−4.33 − 3.26i)12-s + (−2.59 − 0.593i)13-s + (−0.982 + 0.561i)14-s + (−2.48 + 7.09i)15-s + (0.220 + 3.99i)16-s + (4.12 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.369 − 0.929i)2-s + (1.55 − 0.175i)3-s + (−0.726 − 0.687i)4-s + (−0.537 + 1.11i)5-s + (0.413 − 1.51i)6-s + (−0.236 − 0.188i)7-s + (−0.907 + 0.420i)8-s + (1.42 − 0.324i)9-s + (0.837 + 0.911i)10-s + (−1.20 + 0.757i)11-s + (−1.25 − 0.943i)12-s + (−0.720 − 0.164i)13-s + (−0.262 + 0.149i)14-s + (−0.641 + 1.83i)15-s + (0.0550 + 0.998i)16-s + (0.999 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34859 - 0.762243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34859 - 0.762243i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.523 + 1.31i)T \) |
| 29 | \( 1 + (-5.26 + 1.11i)T \) |
| good | 3 | \( 1 + (-2.69 + 0.304i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (1.20 - 2.49i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (0.625 + 0.498i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (4.00 - 2.51i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (2.59 + 0.593i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.12 + 4.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.647 + 5.74i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (-1.26 - 2.62i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-0.0981 - 0.280i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (1.42 - 2.27i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (-6.66 - 6.66i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.02 - 0.357i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-1.28 - 2.03i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (12.1 + 5.87i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 - 11.1iT - 59T^{2} \) |
| 61 | \( 1 + (0.603 + 5.35i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (1.16 + 5.10i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.67 + 7.31i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.02 + 1.05i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (4.66 - 7.42i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (7.68 - 6.13i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (1.83 - 0.641i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.198 - 1.75i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−13.47005593051604811902697957083, −12.55390722279793360722610548803, −11.29903850435698892393906895180, −10.11889043303746701562053935932, −9.447188566902373717274468637794, −7.929949562949260263827165549418, −7.11396177570694490864488489664, −4.81407851185886848483973263924, −3.13734091061777563219916580153, −2.65497502558014582116315618910,
3.10011253100111846886596779386, 4.32499672417130599349634818727, 5.67392006457005483409720709402, 7.67553265790408543269175366484, 8.205544106316443199432560046568, 8.943331203889867006740400810961, 10.12220321813385512277188463720, 12.43341152433355325377144504795, 12.77301519321172633247273704786, 14.03128389266880429730665879783
