| L(s) = 1 | + (37.0 − 64.1i)2-s + (−82.0 − 142. i)3-s + (−1.71e3 − 2.97e3i)4-s + (1.67e3 − 2.89e3i)5-s − 1.21e4·6-s + (−2.63e4 + 3.57e4i)7-s − 1.02e5·8-s + (7.51e4 − 1.30e5i)9-s + (−1.23e5 − 2.14e5i)10-s + (−1.78e5 − 3.09e5i)11-s + (−2.81e5 + 4.87e5i)12-s + 2.06e6·13-s + (1.31e6 + 3.01e6i)14-s − 5.47e5·15-s + (−2.77e5 + 4.80e5i)16-s + (1.53e6 + 2.65e6i)17-s + ⋯ |
| L(s) = 1 | + (0.817 − 1.41i)2-s + (−0.194 − 0.337i)3-s + (−0.837 − 1.45i)4-s + (0.239 − 0.414i)5-s − 0.637·6-s + (−0.593 + 0.804i)7-s − 1.10·8-s + (0.424 − 0.734i)9-s + (−0.390 − 0.677i)10-s + (−0.334 − 0.578i)11-s + (−0.326 + 0.565i)12-s + 1.54·13-s + (0.654 + 1.49i)14-s − 0.186·15-s + (−0.0661 + 0.114i)16-s + (0.262 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.500851 - 2.06749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.500851 - 2.06749i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.63e4 - 3.57e4i)T \) |
| good | 2 | \( 1 + (-37.0 + 64.1i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (82.0 + 142. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-1.67e3 + 2.89e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.78e5 + 3.09e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 2.06e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.53e6 - 2.65e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-6.51e6 + 1.12e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.24e7 - 3.88e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.16e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.09e8 - 1.90e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (3.99e7 - 6.91e7i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 9.01e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.79e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (5.34e8 - 9.24e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.00e8 - 3.46e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.26e9 + 3.92e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (8.45e8 - 1.46e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (4.36e9 + 7.56e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 5.61e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-4.25e9 - 7.37e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.02e10 - 1.77e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 2.59e8T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.21e10 - 7.30e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.58e11T + 7.15e21T^{2} \) |
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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
−19.30188241041996120748918079740, −18.15343186320143329047475341067, −15.72295549497333684364096092234, −13.53553524445426914417358488010, −12.60704203579618902578826769447, −11.25515801773805258860699656451, −9.331715543424835972650611940209, −5.75557779012564176219485422284, −3.39677982741824393096801617179, −1.23003783344996850440734332663,
4.14136085106287228727539889662, 6.07385987796194867023527096240, 7.64889778871291995205207992916, 10.36459631439694226662756013118, 13.11203888770836229107953579724, 14.20585613116844155781728149585, 15.88523790838512515336359658605, 16.60004907448733836903922165169, 18.39555654793459034696506875421, 20.71946440070644294290190530287
