| L(s) = 1 | − 16·2-s − 179.·3-s + 256·4-s + 168.·5-s + 2.86e3·6-s − 4.09e3·8-s + 1.24e4·9-s − 2.69e3·10-s + 1.44e4·11-s − 4.59e4·12-s + 1.09e5·13-s − 3.01e4·15-s + 6.55e4·16-s − 3.00e5·17-s − 1.99e5·18-s + 4.91e5·19-s + 4.30e4·20-s − 2.31e5·22-s − 2.27e6·23-s + 7.34e5·24-s − 1.92e6·25-s − 1.74e6·26-s + 1.29e6·27-s − 3.80e6·29-s + 4.82e5·30-s − 7.90e6·31-s − 1.04e6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.27·3-s + 0.5·4-s + 0.120·5-s + 0.903·6-s − 0.353·8-s + 0.633·9-s − 0.0851·10-s + 0.298·11-s − 0.639·12-s + 1.05·13-s − 0.153·15-s + 0.250·16-s − 0.872·17-s − 0.448·18-s + 0.865·19-s + 0.0602·20-s − 0.211·22-s − 1.69·23-s + 0.451·24-s − 0.985·25-s − 0.749·26-s + 0.468·27-s − 0.998·29-s + 0.108·30-s − 1.53·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6639853167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6639853167\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 179.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 168.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.09e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.90e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.11e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.37e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.51e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.90e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.41e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.87e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.96e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.60e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.18e9T + 7.60e17T^{2} \) |
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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
−11.63298116578669143646803911952, −11.19286773214398049068909571869, −10.07546489549270758585054024497, −8.960647619578488814032562710493, −7.60476078478361955057458244777, −6.30682744844099212666384837525, −5.60398087032197822043428974427, −3.89886169909681029966768748900, −1.86770960949479169947909040124, −0.53912756613642600097762814203,
0.53912756613642600097762814203, 1.86770960949479169947909040124, 3.89886169909681029966768748900, 5.60398087032197822043428974427, 6.30682744844099212666384837525, 7.60476078478361955057458244777, 8.960647619578488814032562710493, 10.07546489549270758585054024497, 11.19286773214398049068909571869, 11.63298116578669143646803911952
