| L(s) = 1 | + (−9.50 − 16.4i)2-s + (−19.7 + 34.1i)3-s + (−116. + 202. i)4-s + (−175. − 303. i)5-s + 750.·6-s + (−666. − 615. i)7-s + 2.01e3·8-s + (314. + 545. i)9-s + (−3.32e3 + 5.76e3i)10-s + (3.13e3 − 5.42e3i)11-s + (−4.61e3 − 7.98e3i)12-s − 4.12e3·13-s + (−3.80e3 + 1.68e4i)14-s + 1.38e4·15-s + (−4.16e3 − 7.21e3i)16-s + (−5.57e3 + 9.64e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.840 − 1.45i)2-s + (−0.421 + 0.730i)3-s + (−0.913 + 1.58i)4-s + (−0.626 − 1.08i)5-s + 1.41·6-s + (−0.734 − 0.678i)7-s + 1.38·8-s + (0.143 + 0.249i)9-s + (−1.05 + 1.82i)10-s + (0.710 − 1.22i)11-s + (−0.770 − 1.33i)12-s − 0.520·13-s + (−0.370 + 1.63i)14-s + 1.05·15-s + (−0.254 − 0.440i)16-s + (−0.275 + 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0521988 + 0.362892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0521988 + 0.362892i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (666. + 615. i)T \) |
| good | 2 | \( 1 + (9.50 + 16.4i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (19.7 - 34.1i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (175. + 303. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-3.13e3 + 5.42e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + 4.12e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (5.57e3 - 9.64e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (8.43e3 + 1.46e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.73e4 + 3.01e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 4.08e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.19e4 + 3.80e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (3.89e4 + 6.75e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 4.18e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.50e5 + 9.53e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-7.80e5 + 1.35e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.37e5 + 2.37e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.52e6 - 2.64e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.22e5 + 5.59e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.31e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (2.32e6 - 4.03e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.17e6 + 2.03e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 1.07e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.98e6 + 6.89e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.44e7T + 8.07e13T^{2} \) |
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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
−19.92294156282556377947785409968, −19.25893378340486019719541051443, −17.06382752959314435093838274744, −16.23897072944535967760156737494, −13.03043506364469910153632753917, −11.53831314497040458516515385387, −10.19056194920516039579594556057, −8.671328102611215181578249686063, −4.03874646303465136081791999022, −0.44749237466508587985111693588,
6.43856025293074487795885380854, 7.34761184331118664116665320680, 9.604588490749770083353432789985, 12.15189993268183297709237712322, 14.70711981823889644161820587131, 15.67014407002093555902792163396, 17.42880321626454532310545391437, 18.41486999469963384756297115307, 19.35631086107609521282484263353, 22.52820531878146363877786210951
