| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 11-s + 13-s + 4·14-s + 16-s − 6·17-s + 8·19-s + 20-s + 22-s + 23-s + 25-s − 26-s − 4·28-s − 6·29-s − 4·31-s − 32-s + 6·34-s − 4·35-s + 2·37-s − 8·38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.223·20-s + 0.213·22-s + 0.208·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.676·35-s + 0.328·37-s − 1.29·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.86378600856573, −12.67310311002259, −11.85335118912662, −11.53187865935736, −11.00846537182635, −10.56121851520163, −10.06203823268548, −9.644255458295659, −9.311532656692207, −8.841767006620731, −8.612389000963547, −7.572160315380617, −7.379680004945348, −6.989407801895050, −6.374296245363764, −5.870751815195107, −5.637116404497762, −4.931304312287294, −4.198290534012836, −3.577648747564492, −3.186828284212310, −2.519682577491375, −2.175988563787500, −1.312060066878912, −0.6711790244262073, 0,
0.6711790244262073, 1.312060066878912, 2.175988563787500, 2.519682577491375, 3.186828284212310, 3.577648747564492, 4.198290534012836, 4.931304312287294, 5.637116404497762, 5.870751815195107, 6.374296245363764, 6.989407801895050, 7.379680004945348, 7.572160315380617, 8.612389000963547, 8.841767006620731, 9.311532656692207, 9.644255458295659, 10.06203823268548, 10.56121851520163, 11.00846537182635, 11.53187865935736, 11.85335118912662, 12.67310311002259, 12.86378600856573
