| L(s) = 1 | + 5.52·2-s − 1.63·3-s + 22.5·4-s + 20.9·5-s − 9.04·6-s − 4.39·7-s + 80.5·8-s − 24.3·9-s + 115.·10-s − 43.1·11-s − 36.9·12-s + 52.6·13-s − 24.3·14-s − 34.2·15-s + 264.·16-s − 50.3·17-s − 134.·18-s + 117.·19-s + 473.·20-s + 7.19·21-s − 238.·22-s − 23·23-s − 131.·24-s + 314.·25-s + 290.·26-s + 83.9·27-s − 99.2·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.314·3-s + 2.82·4-s + 1.87·5-s − 0.615·6-s − 0.237·7-s + 3.56·8-s − 0.901·9-s + 3.66·10-s − 1.18·11-s − 0.887·12-s + 1.12·13-s − 0.464·14-s − 0.589·15-s + 4.14·16-s − 0.718·17-s − 1.76·18-s + 1.42·19-s + 5.29·20-s + 0.0747·21-s − 2.30·22-s − 0.208·23-s − 1.12·24-s + 2.51·25-s + 2.19·26-s + 0.598·27-s − 0.670·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.548584855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.548584855\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.52T + 8T^{2} \) |
| 3 | \( 1 + 1.63T + 27T^{2} \) |
| 5 | \( 1 - 20.9T + 125T^{2} \) |
| 7 | \( 1 + 4.39T + 343T^{2} \) |
| 11 | \( 1 + 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 430.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 553.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 505.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 132.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 646.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 610.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 896.T + 9.12e5T^{2} \) |
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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
−10.54677022256982822824506492378, −9.557973258074731033422625351928, −8.160115799166040553502351264601, −6.74345744833906339048771306552, −6.20422250156188214210817724516, −5.33625991530609754768555144356, −5.14225768233726070604534700485, −3.39756652112911976424335530560, −2.61545116140957090846596533411, −1.60811422901013196864397914965,
1.60811422901013196864397914965, 2.61545116140957090846596533411, 3.39756652112911976424335530560, 5.14225768233726070604534700485, 5.33625991530609754768555144356, 6.20422250156188214210817724516, 6.74345744833906339048771306552, 8.160115799166040553502351264601, 9.557973258074731033422625351928, 10.54677022256982822824506492378
