L(s) = 1 | + 2-s − 3-s + 4-s − 2.09·5-s − 6-s + 0.639·7-s + 8-s + 9-s − 2.09·10-s − 0.777·11-s − 12-s + 0.854·13-s + 0.639·14-s + 2.09·15-s + 16-s + 17-s + 18-s − 4.55·19-s − 2.09·20-s − 0.639·21-s − 0.777·22-s + 7.38·23-s − 24-s − 0.596·25-s + 0.854·26-s − 27-s + 0.639·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.938·5-s − 0.408·6-s + 0.241·7-s + 0.353·8-s + 0.333·9-s − 0.663·10-s − 0.234·11-s − 0.288·12-s + 0.236·13-s + 0.171·14-s + 0.541·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.04·19-s − 0.469·20-s − 0.139·21-s − 0.165·22-s + 1.53·23-s − 0.204·24-s − 0.119·25-s + 0.167·26-s − 0.192·27-s + 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 - 0.639T + 7T^{2} \) |
| 11 | \( 1 + 0.777T + 11T^{2} \) |
| 13 | \( 1 - 0.854T + 13T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 - 0.346T + 31T^{2} \) |
| 37 | \( 1 + 0.563T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 61 | \( 1 - 3.66T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 0.913T + 83T^{2} \) |
| 89 | \( 1 - 1.57T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−7.62398005709127261363334633583, −6.90739658036393031260163801919, −6.31328281202625447846603427265, −5.38594187007478438975384666809, −4.89397817268831239500915826042, −4.04734523289391101206460737714, −3.50962379443007170510115943909, −2.45372598626625116532830422190, −1.32202227175899287975485748119, 0,
1.32202227175899287975485748119, 2.45372598626625116532830422190, 3.50962379443007170510115943909, 4.04734523289391101206460737714, 4.89397817268831239500915826042, 5.38594187007478438975384666809, 6.31328281202625447846603427265, 6.90739658036393031260163801919, 7.62398005709127261363334633583