| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 2·13-s + 16-s − 17-s + 4·19-s + 6·22-s − 5·25-s + 2·26-s + 4·31-s − 32-s + 34-s − 4·37-s − 4·38-s + 6·41-s + 8·43-s − 6·44-s + 5·50-s − 2·52-s + 6·53-s + 4·61-s − 4·62-s + 64-s + 8·67-s − 68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 1.27·22-s − 25-s + 0.392·26-s + 0.718·31-s − 0.176·32-s + 0.171·34-s − 0.657·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.904·44-s + 0.707·50-s − 0.277·52-s + 0.824·53-s + 0.512·61-s − 0.508·62-s + 1/8·64-s + 0.977·67-s − 0.121·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.19735639155593, −15.81894486354278, −15.47614175415650, −14.84512222482307, −13.96452409245311, −13.68270221440269, −12.80120177746566, −12.50789324029013, −11.65460829521950, −11.25789418312381, −10.44541909239074, −10.15916824343610, −9.545798525295212, −8.907728772945927, −8.160618185897864, −7.663088571939708, −7.318513675843698, −6.456107145824030, −5.623387660943705, −5.240033842136342, −4.391358871370364, −3.443174138102097, −2.580109971514423, −2.200364964798855, −0.9422387846645484, 0,
0.9422387846645484, 2.200364964798855, 2.580109971514423, 3.443174138102097, 4.391358871370364, 5.240033842136342, 5.623387660943705, 6.456107145824030, 7.318513675843698, 7.663088571939708, 8.160618185897864, 8.907728772945927, 9.545798525295212, 10.15916824343610, 10.44541909239074, 11.25789418312381, 11.65460829521950, 12.50789324029013, 12.80120177746566, 13.68270221440269, 13.96452409245311, 14.84512222482307, 15.47614175415650, 15.81894486354278, 16.19735639155593
