| L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.662 + 0.382i)5-s + (−0.499 + 0.866i)9-s + (−1.87 + 1.08i)11-s − 0.317i·13-s − 0.765i·15-s + (2.92 − 1.68i)17-s + (−2.82 + 4.89i)19-s + (−4.52 − 2.61i)23-s + (−2.20 − 3.82i)25-s + 0.999·27-s − 2.58·29-s + (−0.828 − 1.43i)31-s + (1.87 + 1.08i)33-s + (0.707 − 1.22i)37-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.296 + 0.171i)5-s + (−0.166 + 0.288i)9-s + (−0.565 + 0.326i)11-s − 0.0879i·13-s − 0.197i·15-s + (0.709 − 0.409i)17-s + (−0.648 + 1.12i)19-s + (−0.943 − 0.544i)23-s + (−0.441 − 0.764i)25-s + 0.192·27-s − 0.480·29-s + (−0.148 − 0.257i)31-s + (0.326 + 0.188i)33-s + (0.116 − 0.201i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3770223086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3770223086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-0.662 - 0.382i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.87 - 1.08i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.317iT - 13T^{2} \) |
| 17 | \( 1 + (-2.92 + 1.68i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.52 + 2.61i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + (0.828 + 1.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.99iT - 41T^{2} \) |
| 43 | \( 1 + 7.39iT - 43T^{2} \) |
| 47 | \( 1 + (4.82 - 8.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.828 + 1.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.12 - 3.53i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.1 - 5.86i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (5.73 - 3.31i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.7 + 6.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + (9.87 + 5.70i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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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
−8.433671105824047185563868547687, −7.81038855261296874715042939003, −7.15733551525161632821499165240, −6.10796438332133951660582527934, −5.74243850622973202525322252628, −4.68469749762153004528419970680, −3.71363681081690221010165376034, −2.51598801195956417341935249337, −1.70761144953030175460241961677, −0.12662470234639794095010739630,
1.49414263657299692455543235619, 2.75503798796121262137416357004, 3.71033974327540760728754644297, 4.63035549678042220275712125022, 5.46272409765264069984645078082, 6.03484466289796943578391830393, 6.99277014244116560652017666100, 7.918851341458185280568664770046, 8.596213159855358352402802018910, 9.498130556348852512534269537019
