| L(s) = 1 | − 2·4-s − 3·7-s − 3·9-s − 5·11-s + 4·16-s + 7·17-s + 4·23-s + 6·28-s + 6·36-s + 43-s + 10·44-s − 13·47-s + 2·49-s + 15·61-s + 9·63-s − 8·64-s − 14·68-s + 11·73-s + 15·77-s + 9·81-s + 16·83-s − 8·92-s + 15·99-s + 10·101-s − 12·112-s − 21·119-s + ⋯ |
| L(s) = 1 | − 4-s − 1.13·7-s − 9-s − 1.50·11-s + 16-s + 1.69·17-s + 0.834·23-s + 1.13·28-s + 36-s + 0.152·43-s + 1.50·44-s − 1.89·47-s + 2/7·49-s + 1.92·61-s + 1.13·63-s − 64-s − 1.69·68-s + 1.28·73-s + 1.70·77-s + 81-s + 1.75·83-s − 0.834·92-s + 1.50·99-s + 0.995·101-s − 1.13·112-s − 1.92·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.61824713974920993594504764555, −6.63088607414418398663848705247, −5.84015582650017493723925533798, −5.29579516351674176013179187813, −4.85335432134469239130548931207, −3.48854845472212746808461755996, −3.31297016434738025974202284029, −2.43814410977130447009590932247, −0.871520748833572046051686868602, 0,
0.871520748833572046051686868602, 2.43814410977130447009590932247, 3.31297016434738025974202284029, 3.48854845472212746808461755996, 4.85335432134469239130548931207, 5.29579516351674176013179187813, 5.84015582650017493723925533798, 6.63088607414418398663848705247, 7.61824713974920993594504764555
