L(s) = 1 | + (−121.5 − 210. i)3-s + (4.94e3 − 8.57e3i)5-s + (2.91e4 + 3.35e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (2.50e4 + 4.33e4i)11-s + 1.46e6·13-s − 2.40e6·15-s + (3.16e6 + 5.48e6i)17-s + (−1.02e7 + 1.77e7i)19-s + (3.52e6 − 1.02e7i)21-s + (−2.36e7 + 4.08e7i)23-s + (−2.45e7 − 4.25e7i)25-s + 1.43e7·27-s − 9.10e7·29-s + (−4.83e7 − 8.37e7i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.708 − 1.22i)5-s + (0.655 + 0.754i)7-s + (−0.166 + 0.288i)9-s + (0.0468 + 0.0810i)11-s + 1.09·13-s − 0.817·15-s + (0.541 + 0.937i)17-s + (−0.949 + 1.64i)19-s + (0.188 − 0.545i)21-s + (−0.764 + 1.32i)23-s + (−0.503 − 0.872i)25-s + 0.192·27-s − 0.824·29-s + (−0.303 − 0.525i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(1.82431 + 0.748466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82431 + 0.748466i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (121.5 + 210. i)T \) |
| 7 | \( 1 + (-2.91e4 - 3.35e4i)T \) |
good | 5 | \( 1 + (-4.94e3 + 8.57e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-2.50e4 - 4.33e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.46e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-3.16e6 - 5.48e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.02e7 - 1.77e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.36e7 - 4.08e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 9.10e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (4.83e7 + 8.37e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (7.14e7 - 1.23e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 8.44e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.55e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (4.22e8 - 7.32e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.26e9 - 2.19e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-2.80e9 - 4.85e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.98e8 - 1.03e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.73e9 + 4.73e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.81e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-9.87e9 - 1.71e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (2.14e10 - 3.71e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.21e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.39e10 - 2.42e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 5.42e10T + 7.15e21T^{2} \) |
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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
−12.35062787955284549691598980455, −11.26893374582447978434665861131, −9.875704402381027854817868844923, −8.643678965339185077309223239978, −7.973468349777581121774718732818, −5.95036703649391333683648726486, −5.57401627012263035820855242948, −3.98874833685868473263797895480, −1.80319303036013705797792586158, −1.36862294757853080611612107831,
0.49994135967724868377117426025, 2.15735372075573587775960526686, 3.49849920653109051172425638090, 4.82533724901993782522448875975, 6.23044347075216826196154195951, 7.10828776304359937668752550390, 8.649534547232917386557250601274, 10.00673832656569447643714616304, 10.81741069955577646393113226741, 11.40670971561633308842081988453