| 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
−14.36100836397543080689764929619, −11.83125835860714985141457038754, −11.17732470108061478110460024509, −10.24402899868171843077147916283, −8.764471267928499359235375485114, −8.155862015656543872843424538357, −5.91058502422663889337167105228, −4.71512894035979707453417871727, −3.92218166243169648709220070539, −1.11409777264849616576783202936,
0.58377829206184513704250286738, 1.39756037806741047995893278943, 2.82421537934753193971941831262, 5.14948330915081587255862247928, 6.93096757424881934589199170994, 7.69638927805045359695479244716, 8.487102559867594840024714574180, 11.01812788345874824107220573053, 11.40280168542028662137973540254, 12.25801272937881123309942472375
