| L(s) = 1 | + (−121.5 − 210. i)3-s + (−1.33e3 + 2.30e3i)5-s + (4.36e4 − 8.49e3i)7-s + (−2.95e4 + 5.11e4i)9-s + (2.62e5 + 4.55e5i)11-s − 1.10e6·13-s + 6.47e5·15-s + (−2.22e5 − 3.84e5i)17-s + (−1.58e6 + 2.74e6i)19-s + (−7.09e6 − 8.15e6i)21-s + (2.38e7 − 4.13e7i)23-s + (2.08e7 + 3.61e7i)25-s + 1.43e7·27-s − 1.65e8·29-s + (−8.05e7 − 1.39e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.190 + 0.330i)5-s + (0.981 − 0.191i)7-s + (−0.166 + 0.288i)9-s + (0.492 + 0.852i)11-s − 0.823·13-s + 0.220·15-s + (−0.0379 − 0.0656i)17-s + (−0.146 + 0.254i)19-s + (−0.378 − 0.435i)21-s + (0.773 − 1.34i)23-s + (0.427 + 0.740i)25-s + 0.192·27-s − 1.49·29-s + (−0.505 − 0.875i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.642178 + 0.831517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642178 + 0.831517i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-4.36e4 + 8.49e3i)T \) |
| good | 5 | \( 1 + (1.33e3 - 2.30e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-2.62e5 - 4.55e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (2.22e5 + 3.84e5i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.58e6 - 2.74e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.38e7 + 4.13e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.65e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (8.05e7 + 1.39e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.73e8 - 3.00e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.07e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.10e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (4.49e7 - 7.77e7i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.97e8 - 3.42e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.65e9 - 6.32e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (6.05e9 - 1.04e10i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-8.63e9 - 1.49e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.31e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (7.39e9 + 1.28e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.45e10 - 2.51e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.43e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.46e10 - 7.73e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 6.63e10T + 7.15e21T^{2} \) |
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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
−12.22662083700093545588202750365, −11.36414097281186980659685960746, −10.35204002193615637218055650157, −8.941080792790587572042324157227, −7.58724060748546376240075850830, −6.91076394029786343635761845584, −5.33725238851227921073030992476, −4.21029978996160614203033559243, −2.40987255273219145174874401213, −1.26723865109850533949483752029,
0.27581885747772363951204043165, 1.69900653811634135627202979332, 3.42757133046747474635865446733, 4.74468358751842810226696184586, 5.62046832331768139328152836740, 7.22109862844833835905374824460, 8.507342326741341678765183251266, 9.403261484895641030169126402242, 10.85235453485381752368283946959, 11.53729592484759083372756631116
