L(s) = 1 | − 10.7·2-s − 1.89·3-s + 84.4·4-s + 25·5-s + 20.4·6-s + 234.·7-s − 566.·8-s − 239.·9-s − 269.·10-s + 121·11-s − 160.·12-s + 279.·13-s − 2.53e3·14-s − 47.4·15-s + 3.40e3·16-s − 1.37e3·17-s + 2.58e3·18-s − 361·19-s + 2.11e3·20-s − 444.·21-s − 1.30e3·22-s − 1.80e3·23-s + 1.07e3·24-s + 625·25-s − 3.01e3·26-s + 914.·27-s + 1.98e4·28-s + ⋯ |
L(s) = 1 | − 1.90·2-s − 0.121·3-s + 2.63·4-s + 0.447·5-s + 0.232·6-s + 1.80·7-s − 3.12·8-s − 0.985·9-s − 0.853·10-s + 0.301·11-s − 0.321·12-s + 0.458·13-s − 3.45·14-s − 0.0544·15-s + 3.32·16-s − 1.15·17-s + 1.87·18-s − 0.229·19-s + 1.18·20-s − 0.220·21-s − 0.575·22-s − 0.710·23-s + 0.380·24-s + 0.200·25-s − 0.874·26-s + 0.241·27-s + 4.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
good | 2 | \( 1 + 10.7T + 32T^{2} \) |
| 3 | \( 1 + 1.89T + 243T^{2} \) |
| 7 | \( 1 - 234.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 279.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.08e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.46e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.23e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.21e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.04e3T + 8.58e9T^{2} \) |
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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
−8.773932254736321610141661648672, −8.179949206406029175762173379837, −7.51614595002082953205460933387, −6.40678308073990105825284403945, −5.75661214051359158883156328042, −4.47712593250652163259696884739, −2.72935146859863878094447970451, −1.90312035513134563007788335648, −1.17054760462917378336864077426, 0,
1.17054760462917378336864077426, 1.90312035513134563007788335648, 2.72935146859863878094447970451, 4.47712593250652163259696884739, 5.75661214051359158883156328042, 6.40678308073990105825284403945, 7.51614595002082953205460933387, 8.179949206406029175762173379837, 8.773932254736321610141661648672