| L(s) = 1 | + 6.96·2-s + 16.4·4-s − 25·5-s + 244.·7-s − 108.·8-s − 174.·10-s − 121·11-s + 742.·13-s + 1.69e3·14-s − 1.27e3·16-s − 170.·17-s + 388.·19-s − 412.·20-s − 842.·22-s − 224.·23-s + 625·25-s + 5.17e3·26-s + 4.02e3·28-s + 5.57e3·29-s − 6.17e3·31-s − 5.45e3·32-s − 1.18e3·34-s − 6.10e3·35-s + 4.51e3·37-s + 2.70e3·38-s + 2.70e3·40-s + 8.77e3·41-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.515·4-s − 0.447·5-s + 1.88·7-s − 0.596·8-s − 0.550·10-s − 0.301·11-s + 1.21·13-s + 2.31·14-s − 1.24·16-s − 0.142·17-s + 0.246·19-s − 0.230·20-s − 0.371·22-s − 0.0885·23-s + 0.200·25-s + 1.50·26-s + 0.969·28-s + 1.23·29-s − 1.15·31-s − 0.941·32-s − 0.175·34-s − 0.841·35-s + 0.542·37-s + 0.303·38-s + 0.266·40-s + 0.815·41-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
| \(L(3)\) |
\(\approx\) |
\(4.706805350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.706805350\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 6.96T + 32T^{2} \) |
| 7 | \( 1 - 244.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 742.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 170.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 388.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 224.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.77e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.81e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.16e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.37e5T + 8.58e9T^{2} \) |
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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
−10.60264251594939121110126989071, −8.987150007365920224778305219591, −8.303372004700780495585986916342, −7.40907435387931412046955595322, −6.10599582038686255203352171878, −5.21532168750058142519687118617, −4.46856428146374591635317476673, −3.65886141438779491833078134327, −2.30540797908668953254207799016, −0.962052022345610837008241556928,
0.962052022345610837008241556928, 2.30540797908668953254207799016, 3.65886141438779491833078134327, 4.46856428146374591635317476673, 5.21532168750058142519687118617, 6.10599582038686255203352171878, 7.40907435387931412046955595322, 8.303372004700780495585986916342, 8.987150007365920224778305219591, 10.60264251594939121110126989071
