| L(s) = 1 | − 3·3-s + 5·5-s + 22·7-s + 9·9-s − 56·11-s + 42·13-s − 15·15-s − 60·17-s − 19·19-s − 66·21-s + 136·23-s + 25·25-s − 27·27-s − 150·29-s + 94·31-s + 168·33-s + 110·35-s + 394·37-s − 126·39-s − 508·41-s − 312·43-s + 45·45-s + 24·47-s + 141·49-s + 180·51-s − 562·53-s − 280·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.18·7-s + 1/3·9-s − 1.53·11-s + 0.896·13-s − 0.258·15-s − 0.856·17-s − 0.229·19-s − 0.685·21-s + 1.23·23-s + 1/5·25-s − 0.192·27-s − 0.960·29-s + 0.544·31-s + 0.886·33-s + 0.531·35-s + 1.75·37-s − 0.517·39-s − 1.93·41-s − 1.10·43-s + 0.149·45-s + 0.0744·47-s + 0.411·49-s + 0.494·51-s − 1.45·53-s − 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
| good | 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 136 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 94 T + p^{3} T^{2} \) |
| 37 | \( 1 - 394 T + p^{3} T^{2} \) |
| 41 | \( 1 + 508 T + p^{3} T^{2} \) |
| 43 | \( 1 + 312 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 562 T + p^{3} T^{2} \) |
| 59 | \( 1 + 554 T + p^{3} T^{2} \) |
| 61 | \( 1 + 350 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 176 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 866 T + p^{3} T^{2} \) |
| 83 | \( 1 - 930 T + p^{3} T^{2} \) |
| 89 | \( 1 + 384 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1206 T + p^{3} T^{2} \) |
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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
−8.221368561597252451859625106067, −7.60726573611170271770589460780, −6.61572138727468186085546381300, −5.86683994656111994939877535326, −4.96303736460717271095404741471, −4.66462293626845639117158127725, −3.25035960324408560115771553228, −2.15060336627532093884686738453, −1.28553546778894417072772187171, 0,
1.28553546778894417072772187171, 2.15060336627532093884686738453, 3.25035960324408560115771553228, 4.66462293626845639117158127725, 4.96303736460717271095404741471, 5.86683994656111994939877535326, 6.61572138727468186085546381300, 7.60726573611170271770589460780, 8.221368561597252451859625106067
