| L(s) = 1 | + (−0.156 − 0.987i)2-s + (0.708 + 0.360i)3-s + (−0.951 + 0.309i)4-s + (1.66 + 1.48i)5-s + (0.245 − 0.755i)6-s + (0.762 − 0.762i)7-s + (0.453 + 0.891i)8-s + (−1.39 − 1.91i)9-s + (1.20 − 1.88i)10-s + (−0.791 − 0.574i)11-s + (−0.784 − 0.124i)12-s + (0.979 − 6.18i)13-s + (−0.872 − 0.633i)14-s + (0.644 + 1.65i)15-s + (0.809 − 0.587i)16-s + (1.89 + 3.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (0.408 + 0.208i)3-s + (−0.475 + 0.154i)4-s + (0.746 + 0.665i)5-s + (0.100 − 0.308i)6-s + (0.288 − 0.288i)7-s + (0.160 + 0.315i)8-s + (−0.464 − 0.638i)9-s + (0.382 − 0.594i)10-s + (−0.238 − 0.173i)11-s + (−0.226 − 0.0358i)12-s + (0.271 − 1.71i)13-s + (−0.233 − 0.169i)14-s + (0.166 + 0.427i)15-s + (0.202 − 0.146i)16-s + (0.460 + 0.904i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59863 - 0.974589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59863 - 0.974589i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 + (-1.66 - 1.48i)T \) |
| 19 | \( 1 + (-4.16 - 1.28i)T \) |
| good | 3 | \( 1 + (-0.708 - 0.360i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.762 + 0.762i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.791 + 0.574i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.979 + 6.18i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 3.72i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (0.781 + 4.93i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.86 + 8.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.51 - 0.493i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.80 - 1.07i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-4.91 - 6.76i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-7.32 - 7.32i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.63 - 3.88i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (7.14 + 3.63i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.00 + 2.18i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (5.44 + 3.95i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.31 + 1.68i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-3.48 + 1.13i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.76 + 11.1i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.74 + 5.36i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (15.3 - 7.82i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (3.80 + 2.76i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.97 - 13.6i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−9.839814256695676248581495094805, −9.445505325034180006889481673610, −8.086308606085278753784818872059, −7.85017133212928147934192442802, −6.14338257171752796944525580088, −5.73914025187316911683865381958, −4.28324671387387335077968814334, −3.18192701130423159511849387526, −2.62695173307328194603833075959, −0.992954698055578465408353049138,
1.44354693971738826723832613507, 2.56205192047181987007006281008, 4.15302114612828018444588009545, 5.29876163060125171828328943799, 5.61597468672168628417503403096, 7.02189181082065333741558830708, 7.55670423229750478945654917687, 8.706935430971384461364952248501, 9.109233396844576238486092845698, 9.766728057648821765269077596525
