| L(s) = 1 | + (−0.949 + 1.04i)2-s + (1.71 − 0.236i)3-s + (−0.196 − 1.99i)4-s + (−0.101 + 2.23i)5-s + (−1.38 + 2.02i)6-s + 0.903·7-s + (2.27 + 1.68i)8-s + (2.88 − 0.811i)9-s + (−2.24 − 2.22i)10-s + (−0.0159 + 0.0115i)11-s + (−0.808 − 3.36i)12-s + (−1.50 + 2.07i)13-s + (−0.857 + 0.946i)14-s + (0.354 + 3.85i)15-s + (−3.92 + 0.783i)16-s + (−0.0389 − 0.119i)17-s + ⋯ |
| L(s) = 1 | + (−0.671 + 0.741i)2-s + (0.990 − 0.136i)3-s + (−0.0983 − 0.995i)4-s + (−0.0452 + 0.998i)5-s + (−0.563 + 0.825i)6-s + 0.341·7-s + (0.803 + 0.595i)8-s + (0.962 − 0.270i)9-s + (−0.709 − 0.704i)10-s + (−0.00480 + 0.00349i)11-s + (−0.233 − 0.972i)12-s + (−0.417 + 0.574i)13-s + (−0.229 + 0.253i)14-s + (0.0916 + 0.995i)15-s + (−0.980 + 0.195i)16-s + (−0.00943 − 0.0290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08307 + 0.751470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08307 + 0.751470i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.949 - 1.04i)T \) |
| 3 | \( 1 + (-1.71 + 0.236i)T \) |
| 5 | \( 1 + (0.101 - 2.23i)T \) |
| good | 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 + (0.0159 - 0.0115i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.50 - 2.07i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0389 + 0.119i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.78 + 1.87i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.34 - 5.98i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (6.74 + 2.19i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.78 - 0.579i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.02 + 2.78i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.58 - 3.55i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + (5.03 + 1.63i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.0726 + 0.223i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (12.0 + 8.78i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.52 + 4.01i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.13 + 12.7i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.59 + 11.0i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.17 + 4.36i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.586 + 0.190i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.0 + 3.58i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (5.84 + 8.04i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 4.00i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.62046301489460868468609439269, −10.82700371970132565087178575597, −9.543366936674507061111127192506, −9.291516067552815254902475299046, −7.73337000932884928918282827277, −7.46615209517645355113403209289, −6.41312764350823698343728609067, −4.97354888106400483003833256712, −3.36594531598397348055033546206, −1.86249414275711634910210670481,
1.32397325185155330779007847654, 2.79465896946104697842151665062, 4.04868844678555677828823757433, 5.15891247790195598109130796377, 7.29513698014703011901744959917, 8.016729701102581663830382398105, 8.856871055446456857757500849311, 9.529886503333071139093062918929, 10.41114910439868002260641508670, 11.53459975788260760644533243513
