| L(s) = 1 | + 2.65e15·2-s + 4.53e30·4-s − 2.85e35·5-s + 3.30e41·7-s + 5.31e45·8-s − 7.60e50·10-s + 4.96e52·11-s + 5.64e55·13-s + 8.78e56·14-s + 2.62e60·16-s + 1.94e61·17-s + 3.52e64·19-s − 1.29e66·20-s + 1.32e68·22-s + 1.63e68·23-s + 4.23e70·25-s + 1.50e71·26-s + 1.49e72·28-s + 5.40e73·29-s − 3.00e75·31-s − 6.47e75·32-s + 5.18e76·34-s − 9.44e76·35-s + 2.08e79·37-s + 9.36e79·38-s − 1.51e81·40-s − 6.19e80·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s − 1.43·5-s + 0.0694·7-s + 1.31·8-s − 2.40·10-s + 1.27·11-s + 0.314·13-s + 0.115·14-s + 0.409·16-s + 0.141·17-s + 0.932·19-s − 2.57·20-s + 2.12·22-s + 0.279·23-s + 1.07·25-s + 0.524·26-s + 0.124·28-s + 0.761·29-s − 1.45·31-s − 0.632·32-s + 0.236·34-s − 0.0999·35-s + 1.33·37-s + 1.55·38-s − 1.89·40-s − 0.222·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(102-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+50.5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(51)\) |
\(\approx\) |
\(5.968523620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.968523620\) |
| \(L(\frac{103}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 2.65e15T + 2.53e30T^{2} \) |
| 5 | \( 1 + 2.85e35T + 3.94e70T^{2} \) |
| 7 | \( 1 - 3.30e41T + 2.26e85T^{2} \) |
| 11 | \( 1 - 4.96e52T + 1.51e105T^{2} \) |
| 13 | \( 1 - 5.64e55T + 3.22e112T^{2} \) |
| 17 | \( 1 - 1.94e61T + 1.88e124T^{2} \) |
| 19 | \( 1 - 3.52e64T + 1.42e129T^{2} \) |
| 23 | \( 1 - 1.63e68T + 3.42e137T^{2} \) |
| 29 | \( 1 - 5.40e73T + 5.03e147T^{2} \) |
| 31 | \( 1 + 3.00e75T + 4.24e150T^{2} \) |
| 37 | \( 1 - 2.08e79T + 2.44e158T^{2} \) |
| 41 | \( 1 + 6.19e80T + 7.78e162T^{2} \) |
| 43 | \( 1 + 1.29e82T + 9.55e164T^{2} \) |
| 47 | \( 1 + 3.59e84T + 7.61e168T^{2} \) |
| 53 | \( 1 - 9.51e86T + 1.41e174T^{2} \) |
| 59 | \( 1 + 2.90e89T + 7.17e178T^{2} \) |
| 61 | \( 1 - 9.18e89T + 2.08e180T^{2} \) |
| 67 | \( 1 + 1.99e92T + 2.71e184T^{2} \) |
| 71 | \( 1 - 1.96e93T + 9.48e186T^{2} \) |
| 73 | \( 1 - 2.27e94T + 1.56e188T^{2} \) |
| 79 | \( 1 - 2.24e95T + 4.57e191T^{2} \) |
| 83 | \( 1 + 1.11e96T + 6.71e193T^{2} \) |
| 89 | \( 1 - 1.83e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 2.50e100T + 4.61e200T^{2} \) |
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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
−9.319393784302207462566527691073, −8.044184581646019713424740836602, −7.08179174609383171850547026145, −6.29255274742225043929179018241, −5.13869605665977909698348229292, −4.31194605541214337652647172104, −3.61850512599943786714870759080, −3.08486226390484758726794464257, −1.69592640987995947825636273847, −0.64570737781554310022841394716,
0.64570737781554310022841394716, 1.69592640987995947825636273847, 3.08486226390484758726794464257, 3.61850512599943786714870759080, 4.31194605541214337652647172104, 5.13869605665977909698348229292, 6.29255274742225043929179018241, 7.08179174609383171850547026145, 8.044184581646019713424740836602, 9.319393784302207462566527691073
