| L(s) = 1 | − 7.43·2-s + 23.2·4-s − 48.3·5-s + 120.·7-s + 65.0·8-s + 359.·10-s + 192.·11-s + 600.·13-s − 895.·14-s − 1.22e3·16-s − 2.19e3·17-s − 393.·19-s − 1.12e3·20-s − 1.42e3·22-s + 212.·23-s − 791.·25-s − 4.46e3·26-s + 2.80e3·28-s + 320.·29-s − 1.19e3·31-s + 7.04e3·32-s + 1.63e4·34-s − 5.82e3·35-s + 1.86e3·37-s + 2.92e3·38-s − 3.14e3·40-s + 1.01e4·41-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.726·4-s − 0.864·5-s + 0.929·7-s + 0.359·8-s + 1.13·10-s + 0.478·11-s + 0.985·13-s − 1.22·14-s − 1.19·16-s − 1.84·17-s − 0.250·19-s − 0.627·20-s − 0.628·22-s + 0.0837·23-s − 0.253·25-s − 1.29·26-s + 0.675·28-s + 0.0707·29-s − 0.223·31-s + 1.21·32-s + 2.42·34-s − 0.803·35-s + 0.224·37-s + 0.328·38-s − 0.310·40-s + 0.939·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 7.43T + 32T^{2} \) |
| 5 | \( 1 + 48.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 120.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 192.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 600.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.19e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 393.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 212.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 320.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.84e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.49e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.53e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.96e4T + 8.58e9T^{2} \) |
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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
−9.312190044753585679046346540280, −8.697938074023668306051917299278, −8.055836013882824986065112308904, −7.26436149292917497762150103971, −6.25796939512266225370310484476, −4.64915085144100436316337877285, −3.95670166184441127598795851064, −2.17075353307151168347399238915, −1.11547355834119626621344761123, 0,
1.11547355834119626621344761123, 2.17075353307151168347399238915, 3.95670166184441127598795851064, 4.64915085144100436316337877285, 6.25796939512266225370310484476, 7.26436149292917497762150103971, 8.055836013882824986065112308904, 8.697938074023668306051917299278, 9.312190044753585679046346540280
