| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 5·11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 5·22-s + 6·23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.06·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146647762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146647762\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.90346491450566, −14.06115412278486, −13.49022677950861, −13.01730597157820, −12.35486822563422, −12.01197861266060, −11.39938523350921, −10.74081168440103, −10.66820017631509, −10.04031781977549, −9.415636589300896, −8.722149227551293, −8.142080673269170, −7.898100130617590, −7.211747307839153, −6.645608926670630, −6.115687911662271, −5.299963855307049, −4.916099892270311, −4.339337906056970, −3.318102245823672, −2.809927431457414, −2.070881149222594, −1.078869751762422, −0.5235930963001398,
0.5235930963001398, 1.078869751762422, 2.070881149222594, 2.809927431457414, 3.318102245823672, 4.339337906056970, 4.916099892270311, 5.299963855307049, 6.115687911662271, 6.645608926670630, 7.211747307839153, 7.898100130617590, 8.142080673269170, 8.722149227551293, 9.415636589300896, 10.04031781977549, 10.66820017631509, 10.74081168440103, 11.39938523350921, 12.01197861266060, 12.35486822563422, 13.01730597157820, 13.49022677950861, 14.06115412278486, 14.90346491450566
