| L(s) = 1 | + (79.7 + 138. i)3-s + (1.01e3 − 1.75e3i)5-s + (4.23e3 + 4.73e3i)7-s + (−2.87e3 + 4.97e3i)9-s + (4.28e3 + 7.42e3i)11-s − 3.84e4·13-s + 3.23e5·15-s + (1.50e5 + 2.61e5i)17-s + (−4.58e5 + 7.93e5i)19-s + (−3.16e5 + 9.62e5i)21-s + (9.66e5 − 1.67e6i)23-s + (−1.08e6 − 1.87e6i)25-s + 2.22e6·27-s + 4.52e6·29-s + (1.47e5 + 2.55e5i)31-s + ⋯ |
| L(s) = 1 | + (0.568 + 0.984i)3-s + (0.725 − 1.25i)5-s + (0.666 + 0.745i)7-s + (−0.146 + 0.252i)9-s + (0.0883 + 0.152i)11-s − 0.373·13-s + 1.64·15-s + (0.438 + 0.759i)17-s + (−0.806 + 1.39i)19-s + (−0.354 + 1.07i)21-s + (0.720 − 1.24i)23-s + (−0.553 − 0.958i)25-s + 0.804·27-s + 1.18·29-s + (0.0286 + 0.0496i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.436210595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436210595\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-4.23e3 - 4.73e3i)T \) |
| good | 3 | \( 1 + (-79.7 - 138. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.01e3 + 1.75e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-4.28e3 - 7.42e3i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 3.84e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.50e5 - 2.61e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.58e5 - 7.93e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-9.66e5 + 1.67e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-1.47e5 - 2.55e5i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.05e6 + 1.39e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.28e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.39e7 - 2.41e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.11e7 - 3.66e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.17e6 - 2.03e6i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.34e6 + 7.52e6i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.26e8 - 2.18e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.32e7 + 4.02e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-5.53e7 + 9.58e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-1.45e8 + 2.52e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.01e8T + 7.60e17T^{2} \) |
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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
−12.27754593334663064637548430939, −10.61369674247681578921113155607, −9.698401224534296744087992554581, −8.810270278119846328449233719803, −8.201624829782707914564137509859, −6.08286730661502503178814646629, −4.97734265766642629005349739392, −4.13499879526438485101741442278, −2.42502259572320575301058280876, −1.18224652161692701641948967692,
0.940602052305639461006447638044, 2.17458848797817144526833985916, 3.06432664821851172174051416177, 4.91967175515559108704470417575, 6.64030255227729630180268272324, 7.15980079613081750544866229844, 8.201292518007336083198156237864, 9.658727917687245309352593928267, 10.71390030860604235833872650493, 11.59674209450351284474772605254
