| L(s) = 1 | + (−20.9 + 36.2i)2-s + (−0.116 − 0.201i)3-s + (−617. − 1.06e3i)4-s + (895. − 1.55e3i)5-s + 9.71·6-s + 3.02e4·8-s + (9.84e3 − 1.70e4i)9-s + (3.74e4 + 6.48e4i)10-s + (−8.70e3 − 1.50e4i)11-s + (−143. + 248. i)12-s − 1.22e5·13-s − 416.·15-s + (−3.15e5 + 5.46e5i)16-s + (−1.65e5 − 2.87e5i)17-s + (4.11e5 + 7.12e5i)18-s + (−3.80e5 + 6.59e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.923 + 1.59i)2-s + (−0.000828 − 0.00143i)3-s + (−1.20 − 2.08i)4-s + (0.641 − 1.11i)5-s + 0.00305·6-s + 2.61·8-s + (0.499 − 0.866i)9-s + (1.18 + 2.05i)10-s + (−0.179 − 0.310i)11-s + (−0.00199 + 0.00346i)12-s − 1.18·13-s − 0.00212·15-s + (−1.20 + 2.08i)16-s + (−0.481 − 0.834i)17-s + (0.923 + 1.59i)18-s + (−0.670 + 1.16i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0343620 - 0.0693163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0343620 - 0.0693163i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (20.9 - 36.2i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (0.116 + 0.201i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-895. + 1.55e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (8.70e3 + 1.50e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.65e5 + 2.87e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (3.80e5 - 6.59e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (6.16e5 - 1.06e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.69e6 - 4.65e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.51e6 + 2.62e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.01e7 - 1.76e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.98e7 - 5.17e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (3.01e7 + 5.22e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.72e6 + 8.17e6i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.09e8 - 1.89e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.27e8 + 3.93e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.25e7 - 3.90e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 3.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.25e8 - 5.64e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.42e9T + 7.60e17T^{2} \) |
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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
−13.62028342457367885774216280744, −12.30992805241088150016715152154, −10.03657642968780903833411187304, −9.357562109289490001009796543693, −8.336647094020639853657419278048, −7.03918035508482039961273687509, −5.81351252137971284106220615182, −4.71788336813818308130224656239, −1.35601632932980676344651627442, −0.03590404328207893094632325832,
2.00837307020022900844561615950, 2.63232809013242700943816071836, 4.46982847097643400897359734487, 6.95117294703796144154877292934, 8.332170058417326107228597780401, 9.881480570620605154336426759887, 10.36861713960893537633099581680, 11.32819752931777709018932199798, 12.68338376026844248388084813765, 13.57838699085757617279592449317
