| L(s) = 1 | − 1.73·2-s − 0.792·3-s + 0.999·4-s + 1.37·6-s − 5.04·7-s + 1.73·8-s − 2.37·9-s + 0.627·11-s − 0.792·12-s − 4.25·13-s + 8.74·14-s − 5·16-s + 1.58·17-s + 4.10·18-s + 4·19-s + 4·21-s − 1.08·22-s − 3.46·23-s − 1.37·24-s + 7.37·26-s + 4.25·27-s − 5.04·28-s − 29-s − 3.37·31-s + 5.19·32-s − 0.497·33-s − 2.74·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.457·3-s + 0.499·4-s + 0.560·6-s − 1.90·7-s + 0.612·8-s − 0.790·9-s + 0.189·11-s − 0.228·12-s − 1.18·13-s + 2.33·14-s − 1.25·16-s + 0.384·17-s + 0.968·18-s + 0.917·19-s + 0.872·21-s − 0.231·22-s − 0.722·23-s − 0.280·24-s + 1.44·26-s + 0.819·27-s − 0.954·28-s − 0.185·29-s − 0.605·31-s + 0.918·32-s − 0.0865·33-s − 0.470·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2843651419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2843651419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 0.792T + 3T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 - 6.74T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 9.80T + 83T^{2} \) |
| 89 | \( 1 + 0.744T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−10.16357660610205429803421057258, −9.466560819225487477253694456281, −9.064054435599781945264339032591, −7.80940631068840242950803479679, −7.07069623654531515843659565612, −6.16863735485524568032576467136, −5.20888133934643334661959641509, −3.68798326217590414048863110178, −2.51384454552368115694380085033, −0.51784198567911311843301615293,
0.51784198567911311843301615293, 2.51384454552368115694380085033, 3.68798326217590414048863110178, 5.20888133934643334661959641509, 6.16863735485524568032576467136, 7.07069623654531515843659565612, 7.80940631068840242950803479679, 9.064054435599781945264339032591, 9.466560819225487477253694456281, 10.16357660610205429803421057258
