| L(s) = 1 | − 7.08·2-s + 18.1·4-s + 79.8·5-s + 97.9·8-s − 565.·10-s − 351.·11-s − 291.·13-s − 1.27e3·16-s + 370.·17-s + 1.50e3·19-s + 1.45e3·20-s + 2.49e3·22-s + 425.·23-s + 3.24e3·25-s + 2.06e3·26-s + 7.78e3·29-s − 2.57e3·31-s + 5.89e3·32-s − 2.62e3·34-s + 739.·37-s − 1.06e4·38-s + 7.82e3·40-s − 7.02e3·41-s + 1.83e3·43-s − 6.39e3·44-s − 3.01e3·46-s + 1.53e3·47-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.567·4-s + 1.42·5-s + 0.541·8-s − 1.78·10-s − 0.876·11-s − 0.478·13-s − 1.24·16-s + 0.310·17-s + 0.956·19-s + 0.810·20-s + 1.09·22-s + 0.167·23-s + 1.03·25-s + 0.599·26-s + 1.71·29-s − 0.481·31-s + 1.01·32-s − 0.388·34-s + 0.0888·37-s − 1.19·38-s + 0.772·40-s − 0.653·41-s + 0.151·43-s − 0.497·44-s − 0.210·46-s + 0.101·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.299992137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299992137\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 7.08T + 32T^{2} \) |
| 5 | \( 1 - 79.8T + 3.12e3T^{2} \) |
| 11 | \( 1 + 351.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 291.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 370.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 425.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 739.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−10.11207979284970771425373509404, −9.539190092735992800108223732176, −8.684564311649365112105163975886, −7.74386408448820548330290241217, −6.83250732722577465365392309973, −5.64423790774398678481654016157, −4.78936294314943478534029878934, −2.84530991157356787320772025739, −1.82013590159465964429512760179, −0.73077884214685122394879259156,
0.73077884214685122394879259156, 1.82013590159465964429512760179, 2.84530991157356787320772025739, 4.78936294314943478534029878934, 5.64423790774398678481654016157, 6.83250732722577465365392309973, 7.74386408448820548330290241217, 8.684564311649365112105163975886, 9.539190092735992800108223732176, 10.11207979284970771425373509404
