| L(s) = 1 | + (−24.1 + 41.8i)3-s + (−904. − 1.56e3i)5-s + (−2.99e3 − 5.60e3i)7-s + (8.67e3 + 1.50e4i)9-s + (1.94e4 − 3.37e4i)11-s + 8.12e4·13-s + 8.74e4·15-s + (2.54e5 − 4.41e5i)17-s + (1.13e5 + 1.96e5i)19-s + (3.06e5 + 9.94e3i)21-s + (−1.22e5 − 2.12e5i)23-s + (−6.59e5 + 1.14e6i)25-s − 1.78e6·27-s + 8.73e5·29-s + (−2.39e6 + 4.15e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.172 + 0.298i)3-s + (−0.647 − 1.12i)5-s + (−0.471 − 0.881i)7-s + (0.440 + 0.763i)9-s + (0.401 − 0.694i)11-s + 0.788·13-s + 0.445·15-s + (0.740 − 1.28i)17-s + (0.199 + 0.346i)19-s + (0.344 + 0.0111i)21-s + (−0.0915 − 0.158i)23-s + (−0.337 + 0.584i)25-s − 0.648·27-s + 0.229·29-s + (−0.466 + 0.808i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.072330779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072330779\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.99e3 + 5.60e3i)T \) |
| good | 3 | \( 1 + (24.1 - 41.8i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (904. + 1.56e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-1.94e4 + 3.37e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 8.12e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-2.54e5 + 4.41e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.13e5 - 1.96e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.22e5 + 2.12e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 8.73e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.39e6 - 4.15e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (7.48e6 + 1.29e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 3.00e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.92e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.01e7 + 3.48e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-3.08e7 + 5.33e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (7.11e7 - 1.23e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.72e7 + 8.17e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.17e8 + 2.03e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.65e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.57e8 - 2.71e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (3.40e7 + 5.89e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.00e6 + 1.73e6i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 4.29e8T + 7.60e17T^{2} \) |
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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
−11.37318118329201986404016654130, −10.38058982267175208184216765419, −9.234681135845377213666734174328, −8.149828516808066741827899295860, −7.11967516759047206371983934981, −5.51959900119684533058721500250, −4.41935029240275060712146714560, −3.45105051850477889028841444562, −1.26420333943874621347418208121, −0.31484535772413564086119823631,
1.44710418866584764018702625718, 3.03187728070302917802578393422, 3.99766049381103810317636675808, 5.96629059966058104824486603065, 6.69913381541301030132682457896, 7.79811025871389841188187319217, 9.161948014889548208553514934489, 10.23448331984795833614618528601, 11.41059759619112877743548946403, 12.20932091482553499832802115144
