| L(s) = 1 | + (−38.1 + 27.7i)2-s + (−502. − 1.54e3i)3-s + (−1.84e3 + 5.67e3i)4-s + (2.96e4 + 2.15e4i)5-s + (6.20e4 + 4.50e4i)6-s + (−2.86e4 + 8.82e4i)7-s + (−2.06e5 − 6.35e5i)8-s + (−8.48e5 + 6.16e5i)9-s − 1.72e6·10-s + (−4.12e6 − 4.18e6i)11-s + 9.70e6·12-s + (1.95e7 − 1.41e7i)13-s + (−1.35e6 − 4.16e6i)14-s + (1.83e7 − 5.66e7i)15-s + (−1.40e7 − 1.02e7i)16-s + (−4.30e7 − 3.12e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.421 + 0.306i)2-s + (−0.397 − 1.22i)3-s + (−0.225 + 0.692i)4-s + (0.847 + 0.615i)5-s + (0.542 + 0.394i)6-s + (−0.0920 + 0.283i)7-s + (−0.278 − 0.856i)8-s + (−0.532 + 0.386i)9-s − 0.546·10-s + (−0.701 − 0.712i)11-s + 0.937·12-s + (1.12 − 0.814i)13-s + (−0.0479 − 0.147i)14-s + (0.416 − 1.28i)15-s + (−0.209 − 0.152i)16-s + (−0.432 − 0.313i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.767420 - 0.676639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.767420 - 0.676639i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.12e6 + 4.18e6i)T \) |
| good | 2 | \( 1 + (38.1 - 27.7i)T + (2.53e3 - 7.79e3i)T^{2} \) |
| 3 | \( 1 + (502. + 1.54e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-2.96e4 - 2.15e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (2.86e4 - 8.82e4i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-1.95e7 + 1.41e7i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (4.30e7 + 3.12e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (8.45e7 + 2.60e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 8.98e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-9.11e7 + 2.80e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (2.22e9 - 1.61e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-4.64e9 + 1.42e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (2.46e9 + 7.58e9i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 6.16e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.88e10 + 5.80e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (1.58e11 - 1.14e11i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-1.28e11 + 3.94e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-2.94e11 - 2.14e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 1.25e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.21e12 + 8.80e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (4.50e11 - 1.38e12i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (6.68e10 - 4.85e10i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (1.00e12 + 7.30e11i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 1.31e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-4.13e12 + 3.00e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−17.48589693991764733364182985658, −15.77221360645382981071054283636, −13.51065093457472546413646870497, −12.85084183970839959984825450711, −10.97877240025896146163073798449, −8.822932389728036636280067675760, −7.25455526803718427384152776947, −6.02398709084623415159922967988, −2.74385253572205843128062078134, −0.57541990787159857220899984585,
1.54881205132222484343665993955, 4.48498395779989262519588422324, 5.78815640114208948541631040775, 8.991015485477183592393106007129, 10.01200876728223124526829618034, 10.97715388171327622158659261673, 13.27859929803475014886144841532, 14.89961275335189529332935994767, 16.30411727581944960967973160959, 17.44851921631279238826482208719
