| L(s) = 1 | + 6·5-s − 20·7-s − 14·11-s − 54·13-s + 66·17-s + 162·19-s + 172·23-s − 89·25-s − 2·29-s + 128·31-s − 120·35-s − 158·37-s − 202·41-s − 298·43-s − 408·47-s + 57·49-s − 690·53-s − 84·55-s + 322·59-s + 298·61-s − 324·65-s + 202·67-s − 700·71-s − 418·73-s + 280·77-s − 744·79-s + 678·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 1.07·7-s − 0.383·11-s − 1.15·13-s + 0.941·17-s + 1.95·19-s + 1.55·23-s − 0.711·25-s − 0.0128·29-s + 0.741·31-s − 0.579·35-s − 0.702·37-s − 0.769·41-s − 1.05·43-s − 1.26·47-s + 0.166·49-s − 1.78·53-s − 0.205·55-s + 0.710·59-s + 0.625·61-s − 0.618·65-s + 0.368·67-s − 1.17·71-s − 0.670·73-s + 0.414·77-s − 1.05·79-s + 0.896·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 162 T + p^{3} T^{2} \) |
| 23 | \( 1 - 172 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 298 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 690 T + p^{3} T^{2} \) |
| 59 | \( 1 - 322 T + p^{3} T^{2} \) |
| 61 | \( 1 - 298 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 + 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 744 T + p^{3} T^{2} \) |
| 83 | \( 1 - 678 T + p^{3} T^{2} \) |
| 89 | \( 1 - 82 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1122 T + p^{3} T^{2} \) |
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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
−9.367819680136505591148583608344, −8.114624921154983130896326857110, −7.26448096331210405248757006681, −6.58226092648848239270279129110, −5.46611035164568730025350378447, −4.97113460198100698672733855639, −3.34625816902438081742851871628, −2.85193507696155299305219449794, −1.37391415515200830909468826826, 0,
1.37391415515200830909468826826, 2.85193507696155299305219449794, 3.34625816902438081742851871628, 4.97113460198100698672733855639, 5.46611035164568730025350378447, 6.58226092648848239270279129110, 7.26448096331210405248757006681, 8.114624921154983130896326857110, 9.367819680136505591148583608344
