| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 0.763·7-s − 8-s + 9-s + 2.85·11-s − 12-s − 2.61·13-s + 0.763·14-s + 16-s + 3.38·17-s − 18-s − 4.47·19-s + 0.763·21-s − 2.85·22-s + 2.38·23-s + 24-s + 2.61·26-s − 27-s − 0.763·28-s + 1.38·29-s + 7.32·31-s − 32-s − 2.85·33-s − 3.38·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.288·7-s − 0.353·8-s + 0.333·9-s + 0.860·11-s − 0.288·12-s − 0.726·13-s + 0.204·14-s + 0.250·16-s + 0.820·17-s − 0.235·18-s − 1.02·19-s + 0.166·21-s − 0.608·22-s + 0.496·23-s + 0.204·24-s + 0.513·26-s − 0.192·27-s − 0.144·28-s + 0.256·29-s + 1.31·31-s − 0.176·32-s − 0.496·33-s − 0.580·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8707754442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8707754442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.763T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 - 2.85T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 3.09T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 7.56T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−10.22082053802855089162175614101, −9.638715378265454509747062731846, −8.726471904841700446386481034339, −7.77993421108922545362757474409, −6.81057986444088451347559307954, −6.19841566373012206493937297684, −5.04380914771904305224511364520, −3.86926482204799545106290735235, −2.44845074914586963366521642903, −0.901396820492146844922788068588,
0.901396820492146844922788068588, 2.44845074914586963366521642903, 3.86926482204799545106290735235, 5.04380914771904305224511364520, 6.19841566373012206493937297684, 6.81057986444088451347559307954, 7.77993421108922545362757474409, 8.726471904841700446386481034339, 9.638715378265454509747062731846, 10.22082053802855089162175614101
