| L(s) = 1 | + (−128. + 223. i)3-s + (125. − 72.2i)5-s + (5.42e3 + 3.31e3i)7-s + (−2.33e4 − 4.05e4i)9-s + (5.95e4 + 3.44e4i)11-s + 9.76e4i·13-s + 3.72e4i·15-s + (−1.90e5 − 1.10e5i)17-s + (−5.49e5 − 9.50e5i)19-s + (−1.43e6 + 7.83e5i)21-s + (−1.68e6 + 9.74e5i)23-s + (−9.66e5 + 1.67e6i)25-s + 6.98e6·27-s − 3.06e6·29-s + (−1.49e6 + 2.58e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.918 + 1.59i)3-s + (0.0895 − 0.0517i)5-s + (0.853 + 0.521i)7-s + (−1.18 − 2.05i)9-s + (1.22 + 0.708i)11-s + 0.948i·13-s + 0.190i·15-s + (−0.554 − 0.320i)17-s + (−0.966 − 1.67i)19-s + (−1.61 + 0.879i)21-s + (−1.25 + 0.726i)23-s + (−0.494 + 0.856i)25-s + 2.53·27-s − 0.804·29-s + (−0.290 + 0.503i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1280274782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1280274782\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5.42e3 - 3.31e3i)T \) |
| good | 3 | \( 1 + (128. - 223. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-125. + 72.2i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-5.95e4 - 3.44e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 9.76e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (1.90e5 + 1.10e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (5.49e5 + 9.50e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.68e6 - 9.74e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.06e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.49e6 - 2.58e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.67e6 - 2.90e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.54e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 2.57e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.26e7 - 2.18e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.75e7 + 4.76e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.63e7 - 2.83e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (9.36e7 - 5.40e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (8.07e7 + 4.66e7i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 4.08e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (2.09e8 + 1.20e8i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.40e8 + 1.96e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 3.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (7.58e8 - 4.37e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 7.10e8iT - 7.60e17T^{2} \) |
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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
−12.05210663848892005422439814036, −11.50896304375587972375093269233, −10.70243797161005906789749181059, −9.290166923538683655600119686989, −9.046979340549957553879734335871, −6.91374153364317786532199143602, −5.68366251962264866112301160046, −4.62448239388600107435778630590, −3.95405129360089769785321395416, −1.89890124653768266119788816213,
0.03882331427854837342493061027, 1.16235596053631441751236042552, 2.03975685637114978340009881475, 4.14966672243618633871370565041, 5.86836290202532070840141958004, 6.35639156910035151467352455386, 7.76431679969006073965413584838, 8.326179315106009835999974211537, 10.38586545876684561533645249028, 11.27962445536848324121858973533
