| L(s) = 1 | + (4 + 6.92i)2-s + (−41 + 71.0i)3-s + (−31.9 + 55.4i)4-s + (224 + 387. i)5-s − 656·6-s − 511.·8-s + (−2.26e3 − 3.92e3i)9-s + (−1.79e3 + 3.10e3i)10-s + (−1.20e3 + 2.08e3i)11-s + (−2.62e3 − 4.54e3i)12-s − 7.11e3·13-s − 3.67e4·15-s + (−2.04e3 − 3.54e3i)16-s + (1.24e3 − 2.15e3i)17-s + (1.81e4 − 3.14e4i)18-s + (1.82e4 + 3.15e4i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.876 + 1.51i)3-s + (−0.249 + 0.433i)4-s + (0.801 + 1.38i)5-s − 1.23·6-s − 0.353·8-s + (−1.03 − 1.79i)9-s + (−0.566 + 0.981i)10-s + (−0.272 + 0.472i)11-s + (−0.438 − 0.759i)12-s − 0.898·13-s − 2.81·15-s + (−0.125 − 0.216i)16-s + (0.0613 − 0.106i)17-s + (0.733 − 1.27i)18-s + (0.610 + 1.05i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.919358 - 0.699407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.919358 - 0.699407i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (41 - 71.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-224 - 387. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.20e3 - 2.08e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + 7.11e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.24e3 + 2.15e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.82e4 - 3.15e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.44e3 - 1.11e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + 8.80e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.41e5 + 2.44e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.07e5 - 1.85e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.64e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.58e5 - 6.20e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.84e4 + 4.93e4i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.07e6 - 1.86e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.54e6 - 2.67e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.51e6 + 2.62e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.06e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-4.94e5 + 8.56e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.70e6 + 2.95e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 1.51e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-8.74e4 - 1.51e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.35e7T + 8.07e13T^{2} \) |
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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
−13.71720789442631473406257196799, −12.09654555222135561344101433674, −11.04556966892293018148239114911, −10.02144975301279993407640641713, −9.580000511810393724058574099492, −7.48817010044403204893241079587, −6.19612510686472654055054516714, −5.42458837748998954429958543458, −4.15986329505504197080669206365, −2.79634909318588009037044448766,
0.38032429338712925123982072119, 1.23613844359551030167109484403, 2.39674511386797836119128447927, 4.96854211467432074996184161531, 5.58610327581473612632563744507, 6.88360500472537270248354219096, 8.315672704508930245738036361036, 9.548297436095923527476968886677, 10.99152858455901272443890732296, 12.05511929407332950370009917858
