| L(s) = 1 | + (31.0 − 53.8i)2-s + (−141. + 245. i)3-s + (−1.42e3 − 2.46e3i)4-s + (−2.94e3 − 1.03e3i)5-s + (8.79e3 + 1.52e4i)6-s − 2.47e4i·7-s − 1.13e5·8-s + (−1.04e4 − 1.81e4i)9-s + (−1.47e5 + 1.26e5i)10-s − 1.15e5·11-s + 8.04e5·12-s + (−3.65e5 − 6.32e5i)13-s + (−1.33e6 − 7.69e5i)14-s + (6.70e5 − 5.76e5i)15-s + (−2.06e6 + 3.57e6i)16-s + (−5.89e5 − 3.40e5i)17-s + ⋯ |
| L(s) = 1 | + (0.971 − 1.68i)2-s + (−0.582 + 1.00i)3-s + (−1.38 − 2.40i)4-s + (−0.943 − 0.331i)5-s + (1.13 + 1.95i)6-s − 1.47i·7-s − 3.45·8-s + (−0.177 − 0.307i)9-s + (−1.47 + 1.26i)10-s − 0.715·11-s + 3.23·12-s + (−0.983 − 1.70i)13-s + (−2.47 − 1.43i)14-s + (0.883 − 0.758i)15-s + (−1.96 + 3.40i)16-s + (−0.415 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.627455 + 0.100770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627455 + 0.100770i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.94e3 + 1.03e3i)T \) |
| 19 | \( 1 + (-1.16e6 + 2.18e6i)T \) |
| good | 2 | \( 1 + (-31.0 + 53.8i)T + (-512 - 886. i)T^{2} \) |
| 3 | \( 1 + (141. - 245. i)T + (-2.95e4 - 5.11e4i)T^{2} \) |
| 7 | \( 1 + 2.47e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.15e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (3.65e5 + 6.32e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + (5.89e5 + 3.40e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 23 | \( 1 + (2.03e6 - 1.17e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-3.50e6 + 2.02e6i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + 1.80e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 4.75e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (-4.64e7 - 2.68e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (4.15e7 + 2.40e7i)T + (1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-3.08e8 + 1.78e8i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + (-2.15e8 - 3.72e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 + (-4.30e8 - 2.48e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (4.04e8 + 7.00e8i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.05e9 + 1.83e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-2.58e8 - 1.49e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (2.55e8 + 1.47e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-2.75e9 - 1.59e9i)T + (4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + 2.29e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (2.59e9 - 1.49e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-3.99e9 + 6.92e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−10.83592971230536590577157646659, −10.48827480472979481900564994363, −9.556558945973070822424921432671, −7.63227821652376811711806094247, −5.37775331873652549321725565391, −4.66551388341066305667354917382, −3.90735253526178755446983353437, −2.79783449074303297312936954055, −0.69643222442313018109492744839, −0.19485532557592415369543648945,
2.52504601868635955014643511035, 4.15900609431120682497957231806, 5.35904826088531036533953296528, 6.34527290000437384528694595035, 7.14103319296383371092872742373, 7.999014897158072698920856394301, 9.088526905266221658619094894029, 11.83642983012646199561631479961, 12.06981423862754989195842436082, 12.93926690014519681294666406053
