| L(s) = 1 | + (0.278 − 1.38i)2-s + (−1.52 + 0.712i)3-s + (−1.84 − 0.772i)4-s + (−0.0344 + 0.0492i)5-s + (0.562 + 2.31i)6-s + (2.22 − 3.84i)7-s + (−1.58 + 2.34i)8-s + (−0.102 + 0.122i)9-s + (0.0586 + 0.0615i)10-s + (−2.72 − 0.730i)11-s + (3.36 − 0.134i)12-s + (−5.08 − 2.37i)13-s + (−4.71 − 4.15i)14-s + (0.0175 − 0.0997i)15-s + (2.80 + 2.84i)16-s + (−3.77 + 3.16i)17-s + ⋯ |
| L(s) = 1 | + (0.196 − 0.980i)2-s + (−0.881 + 0.411i)3-s + (−0.922 − 0.386i)4-s + (−0.0154 + 0.0220i)5-s + (0.229 + 0.945i)6-s + (0.839 − 1.45i)7-s + (−0.560 + 0.828i)8-s + (−0.0341 + 0.0406i)9-s + (0.0185 + 0.0194i)10-s + (−0.821 − 0.220i)11-s + (0.972 − 0.0387i)12-s + (−1.41 − 0.657i)13-s + (−1.26 − 1.10i)14-s + (0.00454 − 0.0257i)15-s + (0.701 + 0.712i)16-s + (−0.914 + 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0523739 + 0.452650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0523739 + 0.452650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.278 + 1.38i)T \) |
| 19 | \( 1 + (3.94 + 1.85i)T \) |
| good | 3 | \( 1 + (1.52 - 0.712i)T + (1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (0.0344 - 0.0492i)T + (-1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-2.22 + 3.84i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.72 + 0.730i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (5.08 + 2.37i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (3.77 - 3.16i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.348 - 1.97i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.631 + 7.21i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-5.51 + 9.55i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.81 - 1.81i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.822 - 0.299i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.99 + 1.39i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-1.07 + 1.28i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.12 - 4.45i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-13.6 - 1.19i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.59 - 5.14i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (5.76 - 0.504i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (9.27 - 1.63i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.10 + 3.04i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (10.1 + 3.68i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.09 + 0.560i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-13.8 + 5.02i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (8.75 + 10.4i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.17485251723254076561348804830, −10.41820654834920476180057442777, −10.01709043533013498912630661942, −8.399103953960156715828508647808, −7.47952379205356157820588179821, −5.82934289357897334448592631602, −4.78944888986409106915683846350, −4.19889560565416981993671029529, −2.38370486572133820711164682981, −0.33208980011096324207703547982,
2.48659425173461407040148557372, 4.83819665233190673865842495910, 5.15808424085918941095459311974, 6.38204923085823548772198106800, 7.12145318249326128051126514622, 8.419914780260537971285567277263, 8.991443034222355550340379447344, 10.36409710534068464877459113141, 11.69089683724657821121124834520, 12.26375585072530314262888683654
