| L(s) = 1 | − 2-s − 1.12·3-s + 4-s − 5-s + 1.12·6-s + 7-s − 8-s − 1.73·9-s + 10-s − 1.12·12-s − 7.06·13-s − 14-s + 1.12·15-s + 16-s − 3.44·17-s + 1.73·18-s − 3.43·19-s − 20-s − 1.12·21-s + 3.63·23-s + 1.12·24-s + 25-s + 7.06·26-s + 5.32·27-s + 28-s + 8.69·29-s − 1.12·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.649·3-s + 0.5·4-s − 0.447·5-s + 0.459·6-s + 0.377·7-s − 0.353·8-s − 0.578·9-s + 0.316·10-s − 0.324·12-s − 1.96·13-s − 0.267·14-s + 0.290·15-s + 0.250·16-s − 0.835·17-s + 0.409·18-s − 0.788·19-s − 0.223·20-s − 0.245·21-s + 0.758·23-s + 0.229·24-s + 0.200·25-s + 1.38·26-s + 1.02·27-s + 0.188·28-s + 1.61·29-s − 0.205·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.12T + 3T^{2} \) |
| 13 | \( 1 + 7.06T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 0.780T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 0.456T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 6.81T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 - 2.88T + 97T^{2} \) |
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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
−7.43549909956875266196547924233, −6.88015285843268356253736991032, −6.22878237024747872011373594147, −5.28891267956918051322532815677, −4.80876665612412252105260457633, −4.00095282060772048168752819309, −2.65919820798311876204164338615, −2.35397432140423889900980526664, −0.874912103265861320843057207188, 0,
0.874912103265861320843057207188, 2.35397432140423889900980526664, 2.65919820798311876204164338615, 4.00095282060772048168752819309, 4.80876665612412252105260457633, 5.28891267956918051322532815677, 6.22878237024747872011373594147, 6.88015285843268356253736991032, 7.43549909956875266196547924233
