| L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s + 1.32i·7-s − i·8-s + 9-s + 10-s + 4.61i·11-s − 12-s − 1.32·14-s − i·15-s + 16-s − 4·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.499i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.39i·11-s − 0.288·12-s − 0.353·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1560412175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1560412175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 1.32iT - 7T^{2} \) |
| 11 | \( 1 - 4.61iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 37 | \( 1 + 6.80iT - 37T^{2} \) |
| 41 | \( 1 - 4.64iT - 41T^{2} \) |
| 43 | \( 1 + 8.60T + 43T^{2} \) |
| 47 | \( 1 + 9.10iT - 47T^{2} \) |
| 53 | \( 1 - 0.826T + 53T^{2} \) |
| 59 | \( 1 - 3.14iT - 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 - 3.18iT - 67T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.28iT - 73T^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 + 11.8iT - 89T^{2} \) |
| 97 | \( 1 + 8.81iT - 97T^{2} \) |
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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
−7.939338147902216999831440485683, −7.48167109478627600450545210515, −6.60713969350273056993844451894, −5.90922196995822943998356067103, −5.11767960808147771266248592381, −4.25382001350607924060568247286, −3.84077263925242884602965473920, −2.33585773188834107740817692773, −1.83408217525523988661528198548, −0.03658281452149543083127763649,
1.28341642676018351495450407354, 2.31700917376104216958011175215, 3.13004537517148227246754051874, 3.73839600008976020152777580467, 4.49194744523020293610979790204, 5.46097040372008363411039035281, 6.38860151550461360290374980957, 6.98757960674748586247880144218, 8.000279027822236193271465814299, 8.460863489678613247478832508394
