| L(s) = 1 | − 3·3-s + 13.3·5-s + 9·9-s − 1.42·11-s − 38.7·13-s − 40.0·15-s − 27.3·17-s + 65.4·19-s + 2.54·23-s + 52.9·25-s − 27·27-s − 63.7·29-s − 51.9·31-s + 4.27·33-s − 335.·37-s + 116.·39-s − 447.·41-s − 170.·43-s + 120.·45-s + 116.·47-s + 82.0·51-s + 86.3·53-s − 19.0·55-s − 196.·57-s − 380.·59-s + 199.·61-s − 517.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.19·5-s + 0.333·9-s − 0.0390·11-s − 0.827·13-s − 0.688·15-s − 0.390·17-s + 0.790·19-s + 0.0230·23-s + 0.423·25-s − 0.192·27-s − 0.408·29-s − 0.301·31-s + 0.0225·33-s − 1.49·37-s + 0.477·39-s − 1.70·41-s − 0.604·43-s + 0.397·45-s + 0.362·47-s + 0.225·51-s + 0.223·53-s − 0.0466·55-s − 0.456·57-s − 0.840·59-s + 0.419·61-s − 0.986·65-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 13.3T + 125T^{2} \) |
| 11 | \( 1 + 1.42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.54T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 116.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 86.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 380.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 199.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 951.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 830.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 332.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 755.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 15.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 101.T + 9.12e5T^{2} \) |
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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
−9.254555512004426082651524427805, −8.185827220745813221662645139032, −7.07311395084747386650666967213, −6.50675497314069263681047861932, −5.37233903081176296172297453510, −5.11029871405416478283511505132, −3.68690951252763864924660131651, −2.39456186353322551859158932999, −1.47462031569902553606617484185, 0,
1.47462031569902553606617484185, 2.39456186353322551859158932999, 3.68690951252763864924660131651, 5.11029871405416478283511505132, 5.37233903081176296172297453510, 6.50675497314069263681047861932, 7.07311395084747386650666967213, 8.185827220745813221662645139032, 9.254555512004426082651524427805
