| L(s) = 1 | + 256·2-s − 6.56e3·3-s + 6.55e4·4-s − 1.99e5·5-s − 1.67e6·6-s + 2.49e7·7-s + 1.67e7·8-s + 4.30e7·9-s − 5.11e7·10-s + 1.25e8·11-s − 4.29e8·12-s + 4.22e9·13-s + 6.38e9·14-s + 1.30e9·15-s + 4.29e9·16-s + 3.55e10·17-s + 1.10e10·18-s − 6.43e10·19-s − 1.30e10·20-s − 1.63e11·21-s + 3.21e10·22-s − 2.45e11·23-s − 1.10e11·24-s − 7.23e11·25-s + 1.08e12·26-s − 2.82e11·27-s + 1.63e12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.228·5-s − 0.408·6-s + 1.63·7-s + 0.353·8-s + 1/3·9-s − 0.161·10-s + 0.176·11-s − 0.288·12-s + 1.43·13-s + 1.15·14-s + 0.131·15-s + 1/4·16-s + 1.23·17-s + 0.235·18-s − 0.869·19-s − 0.114·20-s − 0.944·21-s + 0.124·22-s − 0.654·23-s − 0.204·24-s − 0.947·25-s + 1.01·26-s − 0.192·27-s + 0.818·28-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
| \(L(9)\) |
\(\approx\) |
\(2.626307912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626307912\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 7986 p^{2} T + p^{17} T^{2} \) |
| 7 | \( 1 - 24959264 T + p^{17} T^{2} \) |
| 11 | \( 1 - 11414220 p T + p^{17} T^{2} \) |
| 13 | \( 1 - 325168886 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 35551782594 T + p^{17} T^{2} \) |
| 19 | \( 1 + 64354589764 T + p^{17} T^{2} \) |
| 23 | \( 1 + 245819296200 T + p^{17} T^{2} \) |
| 29 | \( 1 + 2280393162906 T + p^{17} T^{2} \) |
| 31 | \( 1 - 4349964811688 T + p^{17} T^{2} \) |
| 37 | \( 1 - 20770411877318 T + p^{17} T^{2} \) |
| 41 | \( 1 + 97624823830086 T + p^{17} T^{2} \) |
| 43 | \( 1 - 76137596568644 T + p^{17} T^{2} \) |
| 47 | \( 1 - 296069387010240 T + p^{17} T^{2} \) |
| 53 | \( 1 + 213113313107874 T + p^{17} T^{2} \) |
| 59 | \( 1 + 1776690045107580 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1424434275760450 T + p^{17} T^{2} \) |
| 67 | \( 1 + 1599652965063556 T + p^{17} T^{2} \) |
| 71 | \( 1 - 5439386569413960 T + p^{17} T^{2} \) |
| 73 | \( 1 + 3725056002188662 T + p^{17} T^{2} \) |
| 79 | \( 1 - 10282676957218328 T + p^{17} T^{2} \) |
| 83 | \( 1 + 29457780904474692 T + p^{17} T^{2} \) |
| 89 | \( 1 + 43414503538999302 T + p^{17} T^{2} \) |
| 97 | \( 1 - 34754667389544578 T + p^{17} T^{2} \) |
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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
−18.52085939391538439641570837333, −17.02183268939072047918977246533, −15.34779471322567265311873203979, −13.89980280486336168136804635948, −11.97570903917715082513044001523, −10.86764299262521937314260609563, −7.979967634813342262846089195590, −5.80763963497006141663460887818, −4.17174633802322635042617818798, −1.46107608880176370196461429681,
1.46107608880176370196461429681, 4.17174633802322635042617818798, 5.80763963497006141663460887818, 7.979967634813342262846089195590, 10.86764299262521937314260609563, 11.97570903917715082513044001523, 13.89980280486336168136804635948, 15.34779471322567265311873203979, 17.02183268939072047918977246533, 18.52085939391538439641570837333
