| L(s) = 1 | + (−121.5 − 210. i)3-s + (5.57e3 − 9.66e3i)5-s + (2.49e4 − 3.68e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (−1.96e5 − 3.39e5i)11-s − 2.28e6·13-s − 2.71e6·15-s + (−2.79e6 − 4.83e6i)17-s + (6.40e6 − 1.11e7i)19-s + (−1.07e7 − 7.70e5i)21-s + (−1.12e7 + 1.95e7i)23-s + (−3.78e7 − 6.55e7i)25-s + 1.43e7·27-s + 1.65e8·29-s + (9.99e7 + 1.73e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.798 − 1.38i)5-s + (0.560 − 0.828i)7-s + (−0.166 + 0.288i)9-s + (−0.367 − 0.636i)11-s − 1.70·13-s − 0.921·15-s + (−0.476 − 0.826i)17-s + (0.593 − 1.02i)19-s + (−0.575 − 0.0411i)21-s + (−0.365 + 0.633i)23-s + (−0.774 − 1.34i)25-s + 0.192·27-s + 1.49·29-s + (0.626 + 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.364885 + 1.34033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364885 + 1.34033i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-2.49e4 + 3.68e4i)T \) |
| good | 5 | \( 1 + (-5.57e3 + 9.66e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.96e5 + 3.39e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 2.28e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (2.79e6 + 4.83e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-6.40e6 + 1.11e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.12e7 - 1.95e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.65e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-9.99e7 - 1.73e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.10e8 + 3.64e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 7.64e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.38e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (9.33e7 - 1.61e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.62e9 - 2.81e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-6.84e7 - 1.18e8i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.63e9 + 2.82e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (7.66e9 + 1.32e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.53e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.23e10 - 2.13e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.05e10 + 1.83e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.62e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.87e10 - 3.23e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 7.72e10T + 7.15e21T^{2} \) |
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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
−11.59417307576311994826774959369, −10.21774494215773981010595597389, −9.164637602182030978756531623307, −7.981076516781705850030032988226, −6.86526401101572800948665368297, −5.22327495464696247807220570125, −4.76026390076369986394578510207, −2.52083395450606536514718203344, −1.17362704179130728035748434301, −0.35932909387256616627945699770,
2.01907131584570987610109136058, 2.80673441007537146002463914418, 4.61778869314186366302688801725, 5.76684859004323696491598115167, 6.82353112957433544855030700606, 8.181743642070925968500304036010, 9.942100596848277833914184072202, 10.12960464172122188057777224607, 11.52081619854237391499817395382, 12.42621086531116306882725908157
