| L(s) = 1 | − 5-s + 7-s + 5·11-s + 3·13-s + 17-s + 6·19-s + 6·23-s + 25-s − 9·29-s + 4·31-s − 35-s − 2·37-s + 4·41-s + 10·43-s − 47-s + 49-s + 4·53-s − 5·55-s + 8·59-s + 8·61-s − 3·65-s + 12·67-s + 8·71-s + 2·73-s + 5·77-s − 13·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.50·11-s + 0.832·13-s + 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s + 0.624·41-s + 1.52·43-s − 0.145·47-s + 1/7·49-s + 0.549·53-s − 0.674·55-s + 1.04·59-s + 1.02·61-s − 0.372·65-s + 1.46·67-s + 0.949·71-s + 0.234·73-s + 0.569·77-s − 1.46·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175174822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175174822\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.69783184656433, −15.02353608697712, −14.59063444727714, −14.07068976926074, −13.57738218176476, −12.86698172805902, −12.33326383399583, −11.67272453017164, −11.22745934120620, −11.01282542355476, −9.993305055148394, −9.397450910326643, −9.001204019470878, −8.361604327048434, −7.715477223873229, −7.047892706212398, −6.685029499847165, −5.692474387887181, −5.368404436536322, −4.378935372795699, −3.822489389606947, −3.346285924078085, −2.369538309713850, −1.273903793037980, −0.8846710835310096,
0.8846710835310096, 1.273903793037980, 2.369538309713850, 3.346285924078085, 3.822489389606947, 4.378935372795699, 5.368404436536322, 5.692474387887181, 6.685029499847165, 7.047892706212398, 7.715477223873229, 8.361604327048434, 9.001204019470878, 9.397450910326643, 9.993305055148394, 11.01282542355476, 11.22745934120620, 11.67272453017164, 12.33326383399583, 12.86698172805902, 13.57738218176476, 14.07068976926074, 14.59063444727714, 15.02353608697712, 15.69783184656433
