| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 18-s − 19-s − 20-s + 2·21-s + 8·23-s + 24-s + 25-s + 2·26-s − 27-s − 2·28-s + 6·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.45759126601335, −12.78548303924468, −12.48142495605916, −12.00054329972896, −11.56390090613170, −11.11305854186253, −10.55506709892107, −10.10208360904625, −9.848325202979137, −9.136803565102945, −8.744504229896684, −8.236885254215729, −7.665033600088779, −7.111570209967566, −6.591058720135308, −6.511209208500992, −5.655115765745545, −5.055896403666652, −4.667180854176443, −3.941709000226912, −3.270501467699301, −2.777004010059368, −2.213726252167556, −1.188410592474459, −0.7537277117115162, 0,
0.7537277117115162, 1.188410592474459, 2.213726252167556, 2.777004010059368, 3.270501467699301, 3.941709000226912, 4.667180854176443, 5.055896403666652, 5.655115765745545, 6.511209208500992, 6.591058720135308, 7.111570209967566, 7.665033600088779, 8.236885254215729, 8.744504229896684, 9.136803565102945, 9.848325202979137, 10.10208360904625, 10.55506709892107, 11.11305854186253, 11.56390090613170, 12.00054329972896, 12.48142495605916, 12.78548303924468, 13.45759126601335
