L(s) = 1 | + (−2.73 − 0.708i)2-s + 3.81·3-s + (6.99 + 3.88i)4-s − 9.46·5-s + (−10.4 − 2.70i)6-s + 9.26i·7-s + (−16.4 − 15.5i)8-s − 12.4·9-s + (25.9 + 6.70i)10-s − 30.4·11-s + (26.7 + 14.8i)12-s − 64.5i·13-s + (6.56 − 25.3i)14-s − 36.1·15-s + (33.8 + 54.2i)16-s + 109. i·17-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.250i)2-s + 0.735·3-s + (0.874 + 0.485i)4-s − 0.846·5-s + (−0.711 − 0.184i)6-s + 0.500i·7-s + (−0.725 − 0.688i)8-s − 0.459·9-s + (0.819 + 0.212i)10-s − 0.834·11-s + (0.642 + 0.356i)12-s − 1.37i·13-s + (0.125 − 0.484i)14-s − 0.622·15-s + (0.529 + 0.848i)16-s + 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00172077 + 0.0438505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00172077 + 0.0438505i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.73 + 0.708i)T \) |
| 31 | \( 1 + (157. - 71.6i)T \) |
good | 3 | \( 1 - 3.81T + 27T^{2} \) |
| 5 | \( 1 + 9.46T + 125T^{2} \) |
| 7 | \( 1 - 9.26iT - 343T^{2} \) |
| 11 | \( 1 + 30.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 109. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 151. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 186. iT - 2.43e4T^{2} \) |
| 37 | \( 1 + 17.4iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.18iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 575. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 130. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 479. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 508. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 155. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 936.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 635.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 757. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.78e3T + 9.12e5T^{2} \) |
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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
−12.31181002594182732082662575820, −11.15248187307674922165439866542, −10.33048405295275995863606799507, −8.919361944005076609799500396384, −8.236350365289670525572125177223, −7.50049144686758624521394134093, −5.76321574578731100999095154870, −3.55950501681337026374478638298, −2.41985083738496789499615059591, −0.02605072450529109109969541923,
2.25683932071557878392937606226, 3.93280041245336212179896721255, 5.92643117056022873047418529796, 7.57018211093071939640098307105, 7.899530330480007712708982632250, 9.158170043106732365647456973693, 10.06592034712414764296833479104, 11.37681695940347563387454353503, 12.03231207380130906685605299481, 13.91308378892025911402665594805