| L(s) = 1 | + 3-s − 5-s − 2·7-s − 2·9-s + 4·11-s − 13-s − 15-s + 4·19-s − 2·21-s − 23-s + 25-s − 5·27-s + 7·29-s − 7·31-s + 4·33-s + 2·35-s + 4·37-s − 39-s + 3·41-s − 6·43-s + 2·45-s − 13·47-s − 3·49-s − 10·53-s − 4·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.29·29-s − 1.25·31-s + 0.696·33-s + 0.338·35-s + 0.657·37-s − 0.160·39-s + 0.468·41-s − 0.914·43-s + 0.298·45-s − 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.71063408375395939308821748960, −6.73102080244819017386355921436, −6.40935477229377881403943792274, −5.44143623003953362474001114493, −4.64700599053745727690123410905, −3.61383197772591189051377594869, −3.33469446734602157165360726382, −2.42388957128345938101894065388, −1.28108361829890287732195514879, 0,
1.28108361829890287732195514879, 2.42388957128345938101894065388, 3.33469446734602157165360726382, 3.61383197772591189051377594869, 4.64700599053745727690123410905, 5.44143623003953362474001114493, 6.40935477229377881403943792274, 6.73102080244819017386355921436, 7.71063408375395939308821748960
