| L(s) = 1 | − 0.183·2-s − 1.96·4-s − 3.71·5-s + 3.65·7-s + 0.728·8-s + 0.682·10-s + 0.912·11-s − 1.27·13-s − 0.670·14-s + 3.79·16-s − 1.65·17-s + 3.74·19-s + 7.30·20-s − 0.167·22-s + 0.248·23-s + 8.80·25-s + 0.234·26-s − 7.17·28-s − 1.93·29-s − 5.44·31-s − 2.15·32-s + 0.304·34-s − 13.5·35-s − 4.41·37-s − 0.688·38-s − 2.70·40-s − 8.64·41-s + ⋯ |
| L(s) = 1 | − 0.129·2-s − 0.983·4-s − 1.66·5-s + 1.37·7-s + 0.257·8-s + 0.215·10-s + 0.275·11-s − 0.354·13-s − 0.179·14-s + 0.949·16-s − 0.402·17-s + 0.859·19-s + 1.63·20-s − 0.0357·22-s + 0.0518·23-s + 1.76·25-s + 0.0460·26-s − 1.35·28-s − 0.359·29-s − 0.977·31-s − 0.380·32-s + 0.0522·34-s − 2.29·35-s − 0.725·37-s − 0.111·38-s − 0.428·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 0.183T + 2T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 - 0.912T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 0.248T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 + 4.41T + 37T^{2} \) |
| 41 | \( 1 + 8.64T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 0.862T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.06T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 0.860T + 97T^{2} \) |
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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
−8.720434030800298743877945526910, −8.617437232515375805561844244315, −7.51590514820417023968646658140, −7.30288707651259762724786398733, −5.54610759127271640552144971698, −4.73901427489088429007292677698, −4.17478672789360753525515776385, −3.28248436848446846157379616094, −1.44488454173911427878795454703, 0,
1.44488454173911427878795454703, 3.28248436848446846157379616094, 4.17478672789360753525515776385, 4.73901427489088429007292677698, 5.54610759127271640552144971698, 7.30288707651259762724786398733, 7.51590514820417023968646658140, 8.617437232515375805561844244315, 8.720434030800298743877945526910
