| L(s) = 1 | − 2-s + 1.16·3-s + 4-s − 5-s − 1.16·6-s + 7-s − 8-s − 1.63·9-s + 10-s + 1.16·12-s + 1.43·13-s − 14-s − 1.16·15-s + 16-s − 7.28·17-s + 1.63·18-s − 4.33·19-s − 20-s + 1.16·21-s + 2.02·23-s − 1.16·24-s + 25-s − 1.43·26-s − 5.41·27-s + 28-s + 9.11·29-s + 1.16·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.675·3-s + 0.5·4-s − 0.447·5-s − 0.477·6-s + 0.377·7-s − 0.353·8-s − 0.543·9-s + 0.316·10-s + 0.337·12-s + 0.397·13-s − 0.267·14-s − 0.302·15-s + 0.250·16-s − 1.76·17-s + 0.384·18-s − 0.994·19-s − 0.223·20-s + 0.255·21-s + 0.422·23-s − 0.238·24-s + 0.200·25-s − 0.281·26-s − 1.04·27-s + 0.188·28-s + 1.69·29-s + 0.213·30-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
| \(L(1)\) |
\(\approx\) |
\(1.273842725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273842725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.16T + 3T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 7.28T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 3.11T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 + 0.705T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 + 4.87T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 6.62T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 - 0.421T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 - 8.34T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.091297961171911561894317380738, −7.18718295272664609771388731496, −6.59558213675607506064534296886, −5.96038416554034779006494068792, −4.82422799870668552835467923649, −4.25139118355938388080449936369, −3.27532200538429636127554623235, −2.54446588906533650367396748174, −1.85108613436538160961602832999, −0.57424432940998752590300254526,
0.57424432940998752590300254526, 1.85108613436538160961602832999, 2.54446588906533650367396748174, 3.27532200538429636127554623235, 4.25139118355938388080449936369, 4.82422799870668552835467923649, 5.96038416554034779006494068792, 6.59558213675607506064534296886, 7.18718295272664609771388731496, 8.091297961171911561894317380738
