| L(s) = 1 | + 21.4·2-s − 53.2·4-s − 9.90e3·7-s − 1.21e4·8-s − 3.64e4·11-s − 1.64e5·13-s − 2.12e5·14-s − 2.32e5·16-s + 8.23e4·17-s − 6.09e5·19-s − 7.80e5·22-s + 1.88e6·23-s − 3.53e6·26-s + 5.27e5·28-s − 3.39e5·29-s + 5.47e5·31-s + 1.22e6·32-s + 1.76e6·34-s − 5.25e6·37-s − 1.30e7·38-s − 2.05e6·41-s + 6.76e6·43-s + 1.94e6·44-s + 4.03e7·46-s + 3.15e7·47-s + 5.77e7·49-s + 8.77e6·52-s + ⋯ |
| L(s) = 1 | + 0.946·2-s − 0.103·4-s − 1.55·7-s − 1.04·8-s − 0.750·11-s − 1.60·13-s − 1.47·14-s − 0.885·16-s + 0.239·17-s − 1.07·19-s − 0.710·22-s + 1.40·23-s − 1.51·26-s + 0.162·28-s − 0.0890·29-s + 0.106·31-s + 0.207·32-s + 0.226·34-s − 0.461·37-s − 1.01·38-s − 0.113·41-s + 0.301·43-s + 0.0780·44-s + 1.33·46-s + 0.942·47-s + 1.43·49-s + 0.166·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7851667324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7851667324\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 21.4T + 512T^{2} \) |
| 7 | \( 1 + 9.90e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.64e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.64e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 8.23e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.88e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.39e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.47e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.25e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.15e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.89e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.77e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.84e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.36e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.87e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.71e8T + 7.60e17T^{2} \) |
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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
−10.53477484509824179709863885514, −9.666803181600104643234761481933, −8.860185915946072387510854445088, −7.35420252572493294902187665682, −6.40449290892922992706593729534, −5.37944682451515417701618384946, −4.45538658299742777874763763708, −3.21859676142999505119156753435, −2.53850703682217272393900620468, −0.33358994979389725211606946508,
0.33358994979389725211606946508, 2.53850703682217272393900620468, 3.21859676142999505119156753435, 4.45538658299742777874763763708, 5.37944682451515417701618384946, 6.40449290892922992706593729534, 7.35420252572493294902187665682, 8.860185915946072387510854445088, 9.666803181600104643234761481933, 10.53477484509824179709863885514
