| L(s) = 1 | − 2·3-s − 3·5-s − 7-s + 9-s + 2·11-s − 3·13-s + 6·15-s + 6·17-s + 2·21-s + 4·25-s + 4·27-s + 29-s + 2·31-s − 4·33-s + 3·35-s + 10·37-s + 6·39-s − 41-s − 2·43-s − 3·45-s + 6·47-s + 49-s − 12·51-s + 9·53-s − 6·55-s − 4·59-s − 15·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.832·13-s + 1.54·15-s + 1.45·17-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.185·29-s + 0.359·31-s − 0.696·33-s + 0.507·35-s + 1.64·37-s + 0.960·39-s − 0.156·41-s − 0.304·43-s − 0.447·45-s + 0.875·47-s + 1/7·49-s − 1.68·51-s + 1.23·53-s − 0.809·55-s − 0.520·59-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.68696475487024, −14.17833319260137, −13.55213518860597, −12.75349979319893, −12.25365233596795, −12.10202158296693, −11.62035257000919, −11.18335097919000, −10.63208641154316, −9.994727168139966, −9.653069114284854, −8.852049146914627, −8.293554651083500, −7.662429601666173, −7.284050917438094, −6.756804569158266, −5.972372963635462, −5.729313638188772, −4.916094764812286, −4.404803243122162, −3.914909521939114, −3.131710940492270, −2.661019117792526, −1.364497835182342, −0.7174532418125572, 0,
0.7174532418125572, 1.364497835182342, 2.661019117792526, 3.131710940492270, 3.914909521939114, 4.404803243122162, 4.916094764812286, 5.729313638188772, 5.972372963635462, 6.756804569158266, 7.284050917438094, 7.662429601666173, 8.293554651083500, 8.852049146914627, 9.653069114284854, 9.994727168139966, 10.63208641154316, 11.18335097919000, 11.62035257000919, 12.10202158296693, 12.25365233596795, 12.75349979319893, 13.55213518860597, 14.17833319260137, 14.68696475487024
