| L(s) = 1 | + 2·3-s − 3·7-s + 9-s − 3·11-s − 4·13-s − 5·17-s − 19-s − 6·21-s − 4·27-s − 2·29-s − 8·31-s − 6·33-s − 10·37-s − 8·39-s + 6·41-s + 7·43-s − 9·47-s + 2·49-s − 10·51-s − 8·53-s − 2·57-s + 14·59-s + 5·61-s − 3·63-s + 6·71-s + 15·73-s + 9·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 1.21·17-s − 0.229·19-s − 1.30·21-s − 0.769·27-s − 0.371·29-s − 1.43·31-s − 1.04·33-s − 1.64·37-s − 1.28·39-s + 0.937·41-s + 1.06·43-s − 1.31·47-s + 2/7·49-s − 1.40·51-s − 1.09·53-s − 0.264·57-s + 1.82·59-s + 0.640·61-s − 0.377·63-s + 0.712·71-s + 1.75·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5893582529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5893582529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.12274246905928, −14.51221934573505, −14.17833818197792, −13.46088314486243, −13.02400516447950, −12.70956429992933, −12.17620334429637, −11.15796662382408, −10.92127842875013, −10.01781429607442, −9.690167694286782, −9.124161490376206, −8.735302288216374, −7.967622905473733, −7.573631590992258, −6.844525703944068, −6.485995783074756, −5.433309582053149, −5.125700531712136, −4.056598737184853, −3.643737269620714, −2.837382743260119, −2.433100506353659, −1.841436972036655, −0.2469902049931420,
0.2469902049931420, 1.841436972036655, 2.433100506353659, 2.837382743260119, 3.643737269620714, 4.056598737184853, 5.125700531712136, 5.433309582053149, 6.485995783074756, 6.844525703944068, 7.573631590992258, 7.967622905473733, 8.735302288216374, 9.124161490376206, 9.690167694286782, 10.01781429607442, 10.92127842875013, 11.15796662382408, 12.17620334429637, 12.70956429992933, 13.02400516447950, 13.46088314486243, 14.17833818197792, 14.51221934573505, 15.12274246905928
