| L(s) = 1 | + 16·2-s − 121.·3-s + 256·4-s − 2.66e3·5-s − 1.94e3·6-s + 4.09e3·8-s − 4.95e3·9-s − 4.26e4·10-s − 1.88e4·11-s − 3.10e4·12-s − 9.60e4·13-s + 3.23e5·15-s + 6.55e4·16-s − 5.73e5·17-s − 7.92e4·18-s − 8.77e4·19-s − 6.82e5·20-s − 3.02e5·22-s − 1.50e6·23-s − 4.97e5·24-s + 5.15e6·25-s − 1.53e6·26-s + 2.98e6·27-s − 1.78e6·29-s + 5.17e6·30-s + 9.66e6·31-s + 1.04e6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.865·3-s + 0.5·4-s − 1.90·5-s − 0.611·6-s + 0.353·8-s − 0.251·9-s − 1.34·10-s − 0.389·11-s − 0.432·12-s − 0.932·13-s + 1.65·15-s + 0.250·16-s − 1.66·17-s − 0.177·18-s − 0.154·19-s − 0.953·20-s − 0.275·22-s − 1.11·23-s − 0.305·24-s + 2.63·25-s − 0.659·26-s + 1.08·27-s − 0.469·29-s + 1.16·30-s + 1.87·31-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
| \(L(5)\) |
\(\approx\) |
\(0.5541112642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5541112642\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 121.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.66e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.88e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.60e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.73e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.77e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.50e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.78e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.66e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.09e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.17e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.60e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.91e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.43e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.50e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.94e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.01e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.65e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.53e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.40e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.72e8T + 7.60e17T^{2} \) |
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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
−11.92147128688527726018350570226, −11.44655776446552892116192051746, −10.50059809539431034757846781667, −8.513928418155574263356652024434, −7.46299720567873583834609178317, −6.39092433888349279453580594269, −4.89193573226115636496663598108, −4.14992697805289216129774698866, −2.68005509307630478600459374424, −0.36809561925808697777940461695,
0.36809561925808697777940461695, 2.68005509307630478600459374424, 4.14992697805289216129774698866, 4.89193573226115636496663598108, 6.39092433888349279453580594269, 7.46299720567873583834609178317, 8.513928418155574263356652024434, 10.50059809539431034757846781667, 11.44655776446552892116192051746, 11.92147128688527726018350570226
