| L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s − 3·9-s + 10-s − 2·11-s + 2·13-s − 3·14-s + 16-s − 17-s + 3·18-s − 19-s − 20-s + 2·22-s + 3·23-s − 4·25-s − 2·26-s + 3·28-s − 6·29-s − 32-s + 34-s − 3·35-s − 3·36-s − 10·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s − 9-s + 0.316·10-s − 0.603·11-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.229·19-s − 0.223·20-s + 0.426·22-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.566·28-s − 1.11·29-s − 0.176·32-s + 0.171·34-s − 0.507·35-s − 1/2·36-s − 1.64·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7355966671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7355966671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.37688563878602, −13.71157596483311, −13.44822636140577, −12.49612946071206, −12.20914018370096, −11.46298021066483, −11.16578642057253, −10.84996635367179, −10.36198425979932, −9.458618577530398, −9.103741729308600, −8.482738829384278, −8.057470304382843, −7.799150344530153, −7.096773441189307, −6.484196090123794, −5.795394324975403, −5.248565784726114, −4.836465278040399, −3.851359877324336, −3.455375511330639, −2.560644227509899, −2.011224193193908, −1.318250848378651, −0.3294587359916726,
0.3294587359916726, 1.318250848378651, 2.011224193193908, 2.560644227509899, 3.455375511330639, 3.851359877324336, 4.836465278040399, 5.248565784726114, 5.795394324975403, 6.484196090123794, 7.096773441189307, 7.799150344530153, 8.057470304382843, 8.482738829384278, 9.103741729308600, 9.458618577530398, 10.36198425979932, 10.84996635367179, 11.16578642057253, 11.46298021066483, 12.20914018370096, 12.49612946071206, 13.44822636140577, 13.71157596483311, 14.37688563878602
