| L(s) = 1 | − 3·3-s − 4.11·5-s + 9·9-s + 17.9·11-s + 23.4·13-s + 12.3·15-s + 76.2·17-s + 35.5·19-s − 40.7·23-s − 108.·25-s − 27·27-s − 178.·29-s + 31.6·31-s − 53.8·33-s − 54.8·37-s − 70.2·39-s − 190.·41-s − 131.·43-s − 37.0·45-s + 199.·47-s − 228.·51-s + 321.·53-s − 73.8·55-s − 106.·57-s + 163.·59-s + 265.·61-s − 96.4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.368·5-s + 0.333·9-s + 0.491·11-s + 0.499·13-s + 0.212·15-s + 1.08·17-s + 0.429·19-s − 0.369·23-s − 0.864·25-s − 0.192·27-s − 1.14·29-s + 0.183·31-s − 0.283·33-s − 0.243·37-s − 0.288·39-s − 0.725·41-s − 0.468·43-s − 0.122·45-s + 0.619·47-s − 0.627·51-s + 0.834·53-s − 0.181·55-s − 0.248·57-s + 0.360·59-s + 0.556·61-s − 0.184·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.539567851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539567851\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.11T + 125T^{2} \) |
| 11 | \( 1 - 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 190.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 131.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 321.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 163.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 10.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 584.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 183.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 47.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 556.T + 9.12e5T^{2} \) |
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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
−9.543600641907311040783618624470, −8.517390678985110198636010741871, −7.69544794713603259787804284300, −6.92263118801407254667097115432, −5.93193658694550236658827524522, −5.29298008941376845191641812522, −4.08632158761759094452867784328, −3.39846815736164214529371856514, −1.82290130031543820500956281915, −0.67330610387098018844290457915,
0.67330610387098018844290457915, 1.82290130031543820500956281915, 3.39846815736164214529371856514, 4.08632158761759094452867784328, 5.29298008941376845191641812522, 5.93193658694550236658827524522, 6.92263118801407254667097115432, 7.69544794713603259787804284300, 8.517390678985110198636010741871, 9.543600641907311040783618624470
