| L(s) = 1 | + (−0.521 − 1.31i)2-s + (1.38 + 2.72i)3-s + (−1.45 + 1.37i)4-s + (−0.550 − 2.16i)5-s + (2.86 − 3.24i)6-s + (−0.707 − 0.707i)7-s + (2.56 + 1.20i)8-s + (−3.74 + 5.15i)9-s + (−2.56 + 1.85i)10-s + (0.474 + 0.653i)11-s + (−5.76 − 2.06i)12-s + (−0.0801 − 0.505i)13-s + (−0.561 + 1.29i)14-s + (5.14 − 4.51i)15-s + (0.244 − 3.99i)16-s + (6.88 + 3.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.368 − 0.929i)2-s + (0.802 + 1.57i)3-s + (−0.728 + 0.685i)4-s + (−0.246 − 0.969i)5-s + (1.16 − 1.32i)6-s + (−0.267 − 0.267i)7-s + (0.905 + 0.424i)8-s + (−1.24 + 1.71i)9-s + (−0.810 + 0.585i)10-s + (0.143 + 0.196i)11-s + (−1.66 − 0.597i)12-s + (−0.0222 − 0.140i)13-s + (−0.149 + 0.346i)14-s + (1.32 − 1.16i)15-s + (0.0611 − 0.998i)16-s + (1.66 + 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20247 + 0.585774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20247 + 0.585774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.521 + 1.31i)T \) |
| 5 | \( 1 + (0.550 + 2.16i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (-1.38 - 2.72i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-0.474 - 0.653i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0801 + 0.505i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-6.88 - 3.50i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.40 - 7.39i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.516 - 3.26i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 1.24i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.71 - 0.882i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.458 - 0.0726i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (1.69 + 1.23i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (3.73 - 3.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.121 - 0.0618i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (10.1 - 5.17i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (2.93 + 2.13i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.80 + 4.94i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.45 + 10.7i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 3.70i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.94 - 1.25i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.04 + 6.28i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.41 - 4.29i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (8.53 + 11.7i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.880 + 1.72i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.17931471445807268132623691976, −9.832750856239169995156757140280, −9.188901052181585422762442853465, −8.099050066451200123707339050489, −7.929556221389605719075287015886, −5.56936378172793215949276764701, −4.75032049893223464649250079444, −3.64675730843114331127974146942, −3.42334231911252977532872637893, −1.56606207310158730087527065090,
0.790722518020214940829089171563, 2.45852926750170201195553081381, 3.43511978042949192337943061817, 5.26075296367228493440508042351, 6.44672360997107344812363857895, 6.89194718413679491490703430622, 7.61185661660479889262014985358, 8.266178057547743373193113864048, 9.196149459312784041797737922278, 9.971695118564911128782907817000
