| L(s) = 1 | + 1.24·2-s − 0.445·4-s + 5-s − 2.24·7-s − 3.04·8-s + 1.24·10-s − 11-s − 2.80·14-s − 2.91·16-s + 6.49·17-s + 2.33·19-s − 0.445·20-s − 1.24·22-s − 0.198·23-s + 25-s + 1.00·28-s − 3.82·29-s + 1.71·31-s + 2.46·32-s + 8.09·34-s − 2.24·35-s + 4.54·37-s + 2.91·38-s − 3.04·40-s − 5.75·41-s − 3.75·43-s + 0.445·44-s + ⋯ |
| L(s) = 1 | + 0.881·2-s − 0.222·4-s + 0.447·5-s − 0.849·7-s − 1.07·8-s + 0.394·10-s − 0.301·11-s − 0.748·14-s − 0.727·16-s + 1.57·17-s + 0.535·19-s − 0.0995·20-s − 0.265·22-s − 0.0412·23-s + 0.200·25-s + 0.188·28-s − 0.711·29-s + 0.307·31-s + 0.436·32-s + 1.38·34-s − 0.379·35-s + 0.746·37-s + 0.472·38-s − 0.482·40-s − 0.898·41-s − 0.572·43-s + 0.0670·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 0.198T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 - 1.97T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6.87T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.57888896113266503125078178847, −6.35426416578736672797753276736, −6.22482485687130175816787021748, −5.21627087256708054364714415820, −4.95790247058889792920447027046, −3.76051361176831871598243034335, −3.31156340511747248736498561745, −2.61570958057102512126135630168, −1.30647509642092252782956777630, 0,
1.30647509642092252782956777630, 2.61570958057102512126135630168, 3.31156340511747248736498561745, 3.76051361176831871598243034335, 4.95790247058889792920447027046, 5.21627087256708054364714415820, 6.22482485687130175816787021748, 6.35426416578736672797753276736, 7.57888896113266503125078178847
