L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 3·7-s + 3·8-s + 9-s + 2·10-s − 11-s + 12-s + 6·13-s − 3·14-s + 2·15-s − 16-s − 3·17-s − 18-s + 2·20-s − 3·21-s + 22-s − 5·23-s − 3·24-s − 25-s − 6·26-s − 27-s − 3·28-s + 7·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.13·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s − 0.801·14-s + 0.516·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.447·20-s − 0.654·21-s + 0.213·22-s − 1.04·23-s − 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.566·28-s + 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8850248972\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8850248972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.26889136012974, −15.97373502839518, −15.53940213925260, −14.68763714831676, −14.07898291037132, −13.60672631631523, −12.90276009289533, −12.33983706237306, −11.49407062757721, −11.13040031641770, −10.77563168918813, −10.11498870480454, −9.214147825531809, −8.724524134022134, −8.071182973264193, −7.796023312279329, −7.103135584143344, −6.067100309323528, −5.605929642179871, −4.531446986113894, −4.304259474616063, −3.639689301859058, −2.242802218465089, −1.303272754468085, −0.5708373046503868,
0.5708373046503868, 1.303272754468085, 2.242802218465089, 3.639689301859058, 4.304259474616063, 4.531446986113894, 5.605929642179871, 6.067100309323528, 7.103135584143344, 7.796023312279329, 8.071182973264193, 8.724524134022134, 9.214147825531809, 10.11498870480454, 10.77563168918813, 11.13040031641770, 11.49407062757721, 12.33983706237306, 12.90276009289533, 13.60672631631523, 14.07898291037132, 14.68763714831676, 15.53940213925260, 15.97373502839518, 16.26889136012974