| L(s) = 1 | − 2.35·2-s + 3.53·4-s + 2.63·5-s − 4.33·7-s − 3.60·8-s − 6.19·10-s + 3.30·11-s − 1.48·13-s + 10.2·14-s + 1.41·16-s + 3.34·17-s + 3.79·19-s + 9.30·20-s − 7.76·22-s + 23-s + 1.94·25-s + 3.50·26-s − 15.3·28-s − 29-s − 4.86·31-s + 3.88·32-s − 7.85·34-s − 11.4·35-s − 5.99·37-s − 8.93·38-s − 9.50·40-s − 6.66·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.76·4-s + 1.17·5-s − 1.63·7-s − 1.27·8-s − 1.96·10-s + 0.995·11-s − 0.413·13-s + 2.72·14-s + 0.353·16-s + 0.810·17-s + 0.871·19-s + 2.08·20-s − 1.65·22-s + 0.208·23-s + 0.388·25-s + 0.686·26-s − 2.89·28-s − 0.185·29-s − 0.873·31-s + 0.686·32-s − 1.34·34-s − 1.93·35-s − 0.984·37-s − 1.44·38-s − 1.50·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 + 4.33T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 3.79T + 19T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 5.99T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 43 | \( 1 - 0.770T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 + 7.57T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 + 4.46T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.77099800175513099815069417599, −6.98729334097013543001389369588, −6.60713585102618581302409740636, −5.93034226125848290702541502633, −5.14940520118516947761769057907, −3.64205256537456750155858504827, −2.98778408943163163439557957442, −1.96487090315499700264099685304, −1.20517186755313421851155520684, 0,
1.20517186755313421851155520684, 1.96487090315499700264099685304, 2.98778408943163163439557957442, 3.64205256537456750155858504827, 5.14940520118516947761769057907, 5.93034226125848290702541502633, 6.60713585102618581302409740636, 6.98729334097013543001389369588, 7.77099800175513099815069417599
