| L(s) = 1 | − 5-s − 3·7-s + 11-s − 6·13-s + 7·17-s − 5·19-s − 6·23-s + 25-s − 5·29-s + 3·31-s + 3·35-s + 3·37-s − 2·41-s − 4·43-s − 2·47-s + 2·49-s + 53-s − 55-s − 10·59-s + 7·61-s + 6·65-s − 8·67-s + 7·71-s + 14·73-s − 3·77-s − 10·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.301·11-s − 1.66·13-s + 1.69·17-s − 1.14·19-s − 1.25·23-s + 1/5·25-s − 0.928·29-s + 0.538·31-s + 0.507·35-s + 0.493·37-s − 0.312·41-s − 0.609·43-s − 0.291·47-s + 2/7·49-s + 0.137·53-s − 0.134·55-s − 1.30·59-s + 0.896·61-s + 0.744·65-s − 0.977·67-s + 0.830·71-s + 1.63·73-s − 0.341·77-s − 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7849318326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7849318326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.79659054358632403515059479268, −7.19471666240095455020214063084, −6.47362116792275956785814758191, −5.85271771545149072102025958727, −5.03217389431509932422361042566, −4.20268577460922729542844409595, −3.50138957579541417896803920029, −2.79065042318290533890986120472, −1.85337295167665800978053385324, −0.41882948018125157200334206531,
0.41882948018125157200334206531, 1.85337295167665800978053385324, 2.79065042318290533890986120472, 3.50138957579541417896803920029, 4.20268577460922729542844409595, 5.03217389431509932422361042566, 5.85271771545149072102025958727, 6.47362116792275956785814758191, 7.19471666240095455020214063084, 7.79659054358632403515059479268
