| L(s) = 1 | + (−121.5 − 210. i)3-s + (−1.33e3 + 2.32e3i)5-s + (−2.80e4 + 3.45e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (−2.39e5 − 4.14e5i)11-s − 1.17e6·13-s + 6.51e5·15-s + (−2.40e6 − 4.17e6i)17-s + (−4.55e6 + 7.88e6i)19-s + (1.06e7 + 1.70e6i)21-s + (−4.21e6 + 7.30e6i)23-s + (2.08e7 + 3.60e7i)25-s + 1.43e7·27-s + 3.65e7·29-s + (−1.20e7 − 2.09e7i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.191 + 0.332i)5-s + (−0.630 + 0.776i)7-s + (−0.166 + 0.288i)9-s + (−0.448 − 0.776i)11-s − 0.874·13-s + 0.221·15-s + (−0.411 − 0.712i)17-s + (−0.421 + 0.730i)19-s + (0.570 + 0.0912i)21-s + (−0.136 + 0.236i)23-s + (0.426 + 0.738i)25-s + 0.192·27-s + 0.331·29-s + (−0.0758 − 0.131i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.936919 - 0.299559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.936919 - 0.299559i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (2.80e4 - 3.45e4i)T \) |
| good | 5 | \( 1 + (1.33e3 - 2.32e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (2.39e5 + 4.14e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.17e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (2.40e6 + 4.17e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (4.55e6 - 7.88e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (4.21e6 - 7.30e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 3.65e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (1.20e7 + 2.09e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.66e8 - 2.88e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 5.86e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.57e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-4.75e8 + 8.24e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (9.09e8 + 1.57e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-8.43e8 - 1.46e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.82e9 + 6.63e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.92e9 + 5.06e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-4.61e9 - 7.99e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-9.94e9 + 1.72e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 4.78e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (9.70e9 - 1.68e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 8.32e10T + 7.15e21T^{2} \) |
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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
−11.99143788315687683842119810773, −10.98327666729907752878106237290, −9.734717396092647418118944699341, −8.509594141951352789157013370364, −7.29028265178429775141137761193, −6.20097779989732211816377917765, −5.11761574789189181603273813296, −3.27065353733881986909512771656, −2.19848597271560645264408205035, −0.42704390650156206891950355563,
0.61357182181329135704878102634, 2.46384640657069877254269844664, 4.04418648804151586905649076267, 4.88937455620651909520757648475, 6.44302623848961521662122648600, 7.54119226123494703418355903316, 8.976926500973895813905923876604, 10.08575954714741442846747630457, 10.80665935757954457824798054983, 12.27865769262502136028403614432
