L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 9.55·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 19.1·10-s − 35.9·11-s − 12·12-s + 6.86·13-s + 14·14-s + 28.6·15-s + 16·16-s − 16.4·17-s + 18·18-s − 27.6·19-s − 38.2·20-s − 21·21-s − 71.8·22-s − 23·23-s − 24·24-s − 33.7·25-s + 13.7·26-s − 27·27-s + 28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.854·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.604·10-s − 0.985·11-s − 0.288·12-s + 0.146·13-s + 0.267·14-s + 0.493·15-s + 0.250·16-s − 0.235·17-s + 0.235·18-s − 0.333·19-s − 0.427·20-s − 0.218·21-s − 0.696·22-s − 0.208·23-s − 0.204·24-s − 0.270·25-s + 0.103·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.004272684\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.004272684\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
good | 5 | \( 1 + 9.55T + 125T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.86T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 307.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 151.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 337.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 744.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 345.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 465.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 956.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 536.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 541.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 352.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 226.T + 9.12e5T^{2} \) |
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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
−10.00603901130118562836365645886, −8.446488715618765435219059020073, −7.958744088872059774805193424441, −6.93814474744904333447427221608, −6.16414980315537608916367697979, −5.04957028401576268301955165190, −4.52257131753370553412233964009, −3.44590392348046326276590869177, −2.26667085556188909894142384992, −0.69280061746187155936582715679,
0.69280061746187155936582715679, 2.26667085556188909894142384992, 3.44590392348046326276590869177, 4.52257131753370553412233964009, 5.04957028401576268301955165190, 6.16414980315537608916367697979, 6.93814474744904333447427221608, 7.958744088872059774805193424441, 8.446488715618765435219059020073, 10.00603901130118562836365645886