| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.372 + 1.71i)3-s + (0.989 − 0.142i)4-s + (−2.13 + 0.649i)5-s + (−0.249 + 1.73i)6-s + (−1.73 + 3.16i)7-s + (0.977 − 0.212i)8-s + (−0.0607 − 0.0277i)9-s + (−2.08 + 0.800i)10-s + (−1.69 + 1.47i)11-s + (−0.124 + 1.74i)12-s + (4.63 − 2.52i)13-s + (−1.50 + 3.28i)14-s + (−0.314 − 3.90i)15-s + (0.959 − 0.281i)16-s + (1.55 + 2.07i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.214 + 0.987i)3-s + (0.494 − 0.0711i)4-s + (−0.956 + 0.290i)5-s + (−0.101 + 0.707i)6-s + (−0.654 + 1.19i)7-s + (0.345 − 0.0751i)8-s + (−0.0202 − 0.00924i)9-s + (−0.660 + 0.253i)10-s + (−0.511 + 0.443i)11-s + (−0.0360 + 0.504i)12-s + (1.28 − 0.701i)13-s + (−0.400 + 0.877i)14-s + (−0.0813 − 1.00i)15-s + (0.239 − 0.0704i)16-s + (0.376 + 0.502i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0800 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.962837 + 1.04321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962837 + 1.04321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (2.13 - 0.649i)T \) |
| 23 | \( 1 + (-4.14 - 2.42i)T \) |
| good | 3 | \( 1 + (0.372 - 1.71i)T + (-2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (1.73 - 3.16i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (1.69 - 1.47i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.63 + 2.52i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.55 - 2.07i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.731 + 5.08i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-4.62 - 0.664i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (7.65 + 4.92i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.06 + 2.86i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-1.83 - 4.02i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-11.7 - 2.56i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (0.0966 + 0.0966i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.42 + 4.05i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.72 + 5.87i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.06 + 9.43i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.697 - 9.75i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (3.94 - 4.55i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (5.36 + 4.01i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (2.52 + 0.742i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.32 - 1.61i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (11.7 - 7.58i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.86 - 2.93i)T + (73.3 + 63.5i)T^{2} \) |
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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
−12.68418265385953475680876579374, −11.26612938862777705475153600164, −10.90270717396841075485126250452, −9.695988833146875962905916083062, −8.621387707662530724814258075166, −7.35454769575152742455535677143, −6.02754883064868638837538575886, −5.05471988071289977358341637861, −3.87802692274916890583888783297, −2.89051065849399679002087635302,
1.07260217403492683666121524092, 3.39616923397210090077025375540, 4.29821269416177341551805379142, 5.93540159666398533917431532878, 6.97301900342961209549157185690, 7.54920277093473954405502371221, 8.729545874454832167416215163859, 10.41831024339823224191150472288, 11.20343825842854080643549667867, 12.26330720514696940959494687138
