| L(s) = 1 | + 4.94·2-s − 7.50·4-s − 25·5-s − 1.89·7-s − 195.·8-s − 123.·10-s − 121·11-s − 378.·13-s − 9.37·14-s − 727.·16-s + 1.08e3·17-s − 3.11e3·19-s + 187.·20-s − 598.·22-s + 3.60e3·23-s + 625·25-s − 1.87e3·26-s + 14.2·28-s − 2.83e3·29-s + 3.70e3·31-s + 2.65e3·32-s + 5.36e3·34-s + 47.3·35-s − 1.86e3·37-s − 1.54e4·38-s + 4.88e3·40-s − 1.17e4·41-s + ⋯ |
| L(s) = 1 | + 0.874·2-s − 0.234·4-s − 0.447·5-s − 0.0146·7-s − 1.08·8-s − 0.391·10-s − 0.301·11-s − 0.620·13-s − 0.0127·14-s − 0.710·16-s + 0.909·17-s − 1.98·19-s + 0.104·20-s − 0.263·22-s + 1.42·23-s + 0.200·25-s − 0.543·26-s + 0.00342·28-s − 0.625·29-s + 0.692·31-s + 0.458·32-s + 0.795·34-s + 0.00653·35-s − 0.224·37-s − 1.73·38-s + 0.483·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.839485370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839485370\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 4.94T + 32T^{2} \) |
| 7 | \( 1 + 1.89T + 1.68e4T^{2} \) |
| 13 | \( 1 + 378.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.87e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.59e4T + 8.58e9T^{2} \) |
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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
−10.26035486612706435668549853031, −9.165857659463797574336694634904, −8.401729757536340951041049741181, −7.32813123126596418244937911242, −6.27043506182645256664546603487, −5.24267975142966379952154278689, −4.45595776873961866592272041443, −3.50455717588670042111337584216, −2.42492265661629901813095490219, −0.58344481078402957514281747430,
0.58344481078402957514281747430, 2.42492265661629901813095490219, 3.50455717588670042111337584216, 4.45595776873961866592272041443, 5.24267975142966379952154278689, 6.27043506182645256664546603487, 7.32813123126596418244937911242, 8.401729757536340951041049741181, 9.165857659463797574336694634904, 10.26035486612706435668549853031
