| L(s) = 1 | + 2.33·2-s + (−1.23 − 1.03i)3-s + 3.44·4-s + (−0.00307 − 0.00532i)5-s + (−2.87 − 2.40i)6-s + (−0.374 + 2.12i)7-s + 3.36·8-s + (−0.0729 − 0.413i)9-s + (−0.00716 − 0.0124i)10-s + (−0.248 − 0.0904i)11-s + (−4.23 − 3.55i)12-s + (−2.69 + 0.980i)13-s + (−0.872 + 4.94i)14-s + (−0.00171 + 0.00972i)15-s + 0.967·16-s + (0.402 + 2.28i)17-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + (−0.710 − 0.596i)3-s + 1.72·4-s + (−0.00137 − 0.00238i)5-s + (−1.17 − 0.983i)6-s + (−0.141 + 0.801i)7-s + 1.19·8-s + (−0.0243 − 0.137i)9-s + (−0.00226 − 0.00392i)10-s + (−0.0749 − 0.0272i)11-s + (−1.22 − 1.02i)12-s + (−0.747 + 0.271i)13-s + (−0.233 + 1.32i)14-s + (−0.000442 + 0.00251i)15-s + 0.241·16-s + (0.0976 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88296 - 0.324255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88296 - 0.324255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 127 | \( 1 + (-0.647 + 11.2i)T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 + (1.23 + 1.03i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.00307 + 0.00532i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.374 - 2.12i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.248 + 0.0904i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.69 - 0.980i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.402 - 2.28i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (1.60 - 2.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.48 + 0.904i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.76 - 1.37i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (6.12 + 5.14i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-8.24 + 6.91i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-9.26 + 7.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 1.00i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (2.66 + 4.60i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.66 - 6.43i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.83 - 10.4i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.94 + 5.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.03 - 11.5i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.498 - 2.82i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.80 - 3.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.35 + 2.81i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.11 - 0.406i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.507 + 0.878i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.517 + 2.93i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.91325132453830291682072051370, −12.50486657735749235349602246845, −11.81453272314955306341471290419, −10.78044375279895764520739901110, −9.085986721405549248249387132345, −7.32319400797123282723307805114, −6.18732231619789885904440704802, −5.57458951141907620230574418276, −4.16551812205950951349801970869, −2.48562570319218572161418571531,
2.99916818668667356863360045469, 4.52222283745515889849258507647, 5.10414678623461466196987087300, 6.40673595985235069483779237131, 7.53504619836822750343327030161, 9.600170028855739444721770432947, 10.84989483807088545338805154991, 11.39008868830904893334548586030, 12.63122806341396811408082805184, 13.34934813235516085355792641691
