| L(s) = 1 | + 5-s − 7-s − 6·11-s − 13-s − 6·17-s − 3·19-s + 4·23-s + 25-s − 2·29-s − 5·31-s − 35-s + 9·37-s − 12·41-s − 9·43-s + 12·47-s − 6·49-s + 2·53-s − 6·55-s − 4·59-s − 2·61-s − 65-s − 12·67-s − 8·71-s − 2·73-s + 6·77-s + 79-s + 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.688·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.898·31-s − 0.169·35-s + 1.47·37-s − 1.87·41-s − 1.37·43-s + 1.75·47-s − 6/7·49-s + 0.274·53-s − 0.809·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.683·77-s + 0.112·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.87092533118867, −13.65794512557024, −13.45215107182206, −13.21700546157809, −12.62961134144017, −12.20968352441066, −11.36362329901216, −10.95193573359931, −10.53880536800242, −10.10479884222418, −9.467933899807762, −9.014311213686110, −8.451013935186263, −7.959167254043018, −7.308874114153273, −6.811374982701508, −6.348575616979425, −5.543720248854463, −5.280313657325216, −4.568839565372202, −4.097632252045543, −3.073340331090471, −2.733821081956792, −2.123360812287036, −1.424653988626860, 0, 0,
1.424653988626860, 2.123360812287036, 2.733821081956792, 3.073340331090471, 4.097632252045543, 4.568839565372202, 5.280313657325216, 5.543720248854463, 6.348575616979425, 6.811374982701508, 7.308874114153273, 7.959167254043018, 8.451013935186263, 9.014311213686110, 9.467933899807762, 10.10479884222418, 10.53880536800242, 10.95193573359931, 11.36362329901216, 12.20968352441066, 12.62961134144017, 13.21700546157809, 13.45215107182206, 13.65794512557024, 14.87092533118867
