| L(s) = 1 | + 6.88·3-s − 5.64·5-s − 7·7-s + 20.4·9-s − 17.3·11-s + 18.8·13-s − 38.8·15-s − 124.·17-s + 153.·19-s − 48.2·21-s + 163.·23-s − 93.1·25-s − 45.1·27-s + 19.9·29-s − 120.·31-s − 119.·33-s + 39.5·35-s − 329.·37-s + 129.·39-s − 41·41-s + 11.0·43-s − 115.·45-s + 80.0·47-s + 49·49-s − 856.·51-s − 564.·53-s + 98.0·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.504·5-s − 0.377·7-s + 0.757·9-s − 0.476·11-s + 0.402·13-s − 0.669·15-s − 1.77·17-s + 1.85·19-s − 0.501·21-s + 1.48·23-s − 0.745·25-s − 0.321·27-s + 0.127·29-s − 0.696·31-s − 0.631·33-s + 0.190·35-s − 1.46·37-s + 0.533·39-s − 0.156·41-s + 0.0391·43-s − 0.382·45-s + 0.248·47-s + 0.142·49-s − 2.35·51-s − 1.46·53-s + 0.240·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 6.88T + 27T^{2} \) |
| 5 | \( 1 + 5.64T + 125T^{2} \) |
| 11 | \( 1 + 17.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 153.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 329.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 11.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 80.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 564.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 679.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 427.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 235.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 125.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 168.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 359.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 2.17T + 7.04e5T^{2} \) |
| 97 | \( 1 + 925.T + 9.12e5T^{2} \) |
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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
−9.101665528269478761238377342534, −8.246342727435086789597976032949, −7.48240171457685803482532504994, −6.82534366003519034804432449948, −5.53128567238790224963073012443, −4.45742087982610658418714006560, −3.40197587396510000607199318810, −2.87417988175999052136814368012, −1.63914466930800954443637796427, 0,
1.63914466930800954443637796427, 2.87417988175999052136814368012, 3.40197587396510000607199318810, 4.45742087982610658418714006560, 5.53128567238790224963073012443, 6.82534366003519034804432449948, 7.48240171457685803482532504994, 8.246342727435086789597976032949, 9.101665528269478761238377342534
