| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 11-s − 12-s + 4·13-s − 2·15-s + 16-s + 17-s + 18-s + 8·19-s + 2·20-s − 22-s − 24-s − 25-s + 4·26-s − 27-s − 2·30-s − 10·31-s + 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.447·20-s − 0.213·22-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.365·30-s − 1.79·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.609701215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.609701215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.42075202754890, −13.81118458919781, −13.34344001704960, −12.93947878890726, −12.61952624336127, −11.68429059338121, −11.44602591893847, −11.02345430990371, −10.35549681833935, −9.792753752863759, −9.442359912298794, −8.807834687263979, −7.810254932563025, −7.667878675214614, −6.817542396878068, −6.259183920476328, −5.814780405331650, −5.382855507052856, −4.923163265837702, −4.121992285293261, −3.434624392969905, −2.989614239471461, −1.995184953364909, −1.493337924749358, −0.7006831328325365,
0.7006831328325365, 1.493337924749358, 1.995184953364909, 2.989614239471461, 3.434624392969905, 4.121992285293261, 4.923163265837702, 5.382855507052856, 5.814780405331650, 6.259183920476328, 6.817542396878068, 7.667878675214614, 7.810254932563025, 8.807834687263979, 9.442359912298794, 9.792753752863759, 10.35549681833935, 11.02345430990371, 11.44602591893847, 11.68429059338121, 12.61952624336127, 12.93947878890726, 13.34344001704960, 13.81118458919781, 14.42075202754890
