| L(s) = 1 | − 256·2-s + 6.56e3·3-s + 6.55e4·4-s − 7.21e4·5-s − 1.67e6·6-s − 8.64e6·7-s − 1.67e7·8-s + 4.30e7·9-s + 1.84e7·10-s + 1.15e9·11-s + 4.29e8·12-s + 2.80e9·13-s + 2.21e9·14-s − 4.73e8·15-s + 4.29e9·16-s + 3.29e10·17-s − 1.10e10·18-s + 5.77e9·19-s − 4.73e9·20-s − 5.66e10·21-s − 2.96e11·22-s + 1.69e11·23-s − 1.10e11·24-s − 7.57e11·25-s − 7.17e11·26-s + 2.82e11·27-s − 5.66e11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.0826·5-s − 0.408·6-s − 0.566·7-s − 0.353·8-s + 1/3·9-s + 0.0584·10-s + 1.63·11-s + 0.288·12-s + 0.952·13-s + 0.400·14-s − 0.0477·15-s + 1/4·16-s + 1.14·17-s − 0.235·18-s + 0.0780·19-s − 0.0413·20-s − 0.327·21-s − 1.15·22-s + 0.450·23-s − 0.204·24-s − 0.993·25-s − 0.673·26-s + 0.192·27-s − 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.628119343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628119343\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{8} T \) |
| 3 | \( 1 - p^{8} T \) |
| good | 5 | \( 1 + 72186 T + p^{17} T^{2} \) |
| 7 | \( 1 + 1234312 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 1159304460 T + p^{17} T^{2} \) |
| 13 | \( 1 - 215466374 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 32979662226 T + p^{17} T^{2} \) |
| 19 | \( 1 - 5778498836 T + p^{17} T^{2} \) |
| 23 | \( 1 - 169116994200 T + p^{17} T^{2} \) |
| 29 | \( 1 - 3631735478814 T + p^{17} T^{2} \) |
| 31 | \( 1 - 6880978560608 T + p^{17} T^{2} \) |
| 37 | \( 1 + 35464500749338 T + p^{17} T^{2} \) |
| 41 | \( 1 + 8923766734806 T + p^{17} T^{2} \) |
| 43 | \( 1 + 129966457018324 T + p^{17} T^{2} \) |
| 47 | \( 1 - 129499777218480 T + p^{17} T^{2} \) |
| 53 | \( 1 - 218262107088054 T + p^{17} T^{2} \) |
| 59 | \( 1 + 1783401246652740 T + p^{17} T^{2} \) |
| 61 | \( 1 - 1469145893932670 T + p^{17} T^{2} \) |
| 67 | \( 1 - 5051560974054596 T + p^{17} T^{2} \) |
| 71 | \( 1 + 793480696785720 T + p^{17} T^{2} \) |
| 73 | \( 1 - 6343500933237962 T + p^{17} T^{2} \) |
| 79 | \( 1 + 8292883305185392 T + p^{17} T^{2} \) |
| 83 | \( 1 + 24031501915598508 T + p^{17} T^{2} \) |
| 89 | \( 1 + 15466463339248422 T + p^{17} T^{2} \) |
| 97 | \( 1 - 79745962551777122 T + p^{17} T^{2} \) |
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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
−18.89588460937458727306980793225, −17.13918254888781484059126801558, −15.68639384633059540137536789705, −13.99549413614453409775290929756, −11.94218690827246809082798237755, −9.900216521220966657910883523337, −8.500803736176823313262696496445, −6.58077774128280343926791868907, −3.46936097626838914562436784176, −1.23171345330008856175470488410,
1.23171345330008856175470488410, 3.46936097626838914562436784176, 6.58077774128280343926791868907, 8.500803736176823313262696496445, 9.900216521220966657910883523337, 11.94218690827246809082798237755, 13.99549413614453409775290929756, 15.68639384633059540137536789705, 17.13918254888781484059126801558, 18.89588460937458727306980793225
