| L(s) = 1 | + (−25.8 − 18.8i)2-s + (−246. − 757. i)3-s + (316. + 973. i)4-s + (−6.03e3 + 3.52e3i)5-s + (−7.87e3 + 2.42e4i)6-s + 7.39e4·7-s + (1.01e4 − 3.11e4i)8-s + (−3.69e5 + 2.68e5i)9-s + (2.22e5 + 2.21e4i)10-s + (7.87e4 + 5.72e4i)11-s + (6.59e5 − 4.79e5i)12-s + (1.11e6 − 8.11e5i)13-s + (−1.91e6 − 1.39e6i)14-s + (4.15e6 + 3.70e6i)15-s + (−8.48e5 + 6.16e5i)16-s + (2.83e6 − 8.72e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.584 − 1.79i)3-s + (0.154 + 0.475i)4-s + (−0.863 + 0.504i)5-s + (−0.413 + 1.27i)6-s + 1.66·7-s + (0.109 − 0.336i)8-s + (−2.08 + 1.51i)9-s + (0.703 + 0.0699i)10-s + (0.147 + 0.107i)11-s + (0.765 − 0.556i)12-s + (0.833 − 0.605i)13-s + (−0.951 − 0.691i)14-s + (1.41 + 1.25i)15-s + (−0.202 + 0.146i)16-s + (0.484 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.177036 - 1.15363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177036 - 1.15363i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 + 18.8i)T \) |
| 5 | \( 1 + (6.03e3 - 3.52e3i)T \) |
| good | 3 | \( 1 + (246. + 757. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 7 | \( 1 - 7.39e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + (-7.87e4 - 5.72e4i)T + (8.81e10 + 2.71e11i)T^{2} \) |
| 13 | \( 1 + (-1.11e6 + 8.11e5i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-2.83e6 + 8.72e6i)T + (-2.77e13 - 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-2.00e6 + 6.17e6i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 + (-1.49e7 - 1.08e7i)T + (2.94e14 + 9.06e14i)T^{2} \) |
| 29 | \( 1 + (-6.04e7 - 1.85e8i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-2.94e7 + 9.06e7i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-2.62e8 + 1.90e8i)T + (5.49e16 - 1.69e17i)T^{2} \) |
| 41 | \( 1 + (3.28e8 - 2.38e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 + 7.85e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.18e8 - 3.65e8i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (1.36e9 + 4.19e9i)T + (-7.49e18 + 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-7.18e8 + 5.22e8i)T + (9.31e18 - 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-6.39e9 - 4.64e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 + (-1.91e9 + 5.89e9i)T + (-9.88e19 - 7.17e19i)T^{2} \) |
| 71 | \( 1 + (1.76e9 + 5.43e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (1.71e10 + 1.24e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (1.96e9 + 6.05e9i)T + (-6.05e20 + 4.39e20i)T^{2} \) |
| 83 | \( 1 + (1.74e10 - 5.37e10i)T + (-1.04e21 - 7.56e20i)T^{2} \) |
| 89 | \( 1 + (-2.96e10 - 2.15e10i)T + (8.57e20 + 2.63e21i)T^{2} \) |
| 97 | \( 1 + (-2.54e9 - 7.84e9i)T + (-5.78e21 + 4.20e21i)T^{2} \) |
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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
−12.25801272937881123309942472375, −11.40280168542028662137973540254, −11.01812788345874824107220573053, −8.487102559867594840024714574180, −7.69638927805045359695479244716, −6.93096757424881934589199170994, −5.14948330915081587255862247928, −2.82421537934753193971941831262, −1.39756037806741047995893278943, −0.58377829206184513704250286738,
1.11409777264849616576783202936, 3.92218166243169648709220070539, 4.71512894035979707453417871727, 5.91058502422663889337167105228, 8.155862015656543872843424538357, 8.764471267928499359235375485114, 10.24402899868171843077147916283, 11.17732470108061478110460024509, 11.83125835860714985141457038754, 14.36100836397543080689764929619
