| L(s) = 1 | + (−48.0 − 83.2i)2-s + (45.1 − 78.1i)3-s + (−519. + 899. i)4-s + (−2.83e4 − 4.90e4i)5-s − 8.66e3·6-s + (5.21e4 + 3.06e5i)7-s − 6.87e5·8-s + (7.93e5 + 1.37e6i)9-s + (−2.72e6 + 4.71e6i)10-s + (4.96e4 − 8.59e4i)11-s + (4.68e4 + 8.11e4i)12-s − 2.66e7·13-s + (2.30e7 − 1.90e7i)14-s − 5.11e6·15-s + (3.72e7 + 6.45e7i)16-s + (2.70e7 − 4.67e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.530 − 0.919i)2-s + (0.0357 − 0.0618i)3-s + (−0.0633 + 0.109i)4-s + (−0.810 − 1.40i)5-s − 0.0758·6-s + (0.167 + 0.985i)7-s − 0.926·8-s + (0.497 + 0.861i)9-s + (−0.860 + 1.49i)10-s + (0.00844 − 0.0146i)11-s + (0.00452 + 0.00784i)12-s − 1.53·13-s + (0.817 − 0.677i)14-s − 0.115·15-s + (0.555 + 0.961i)16-s + (0.271 − 0.469i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.170618 + 0.334993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170618 + 0.334993i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.21e4 - 3.06e5i)T \) |
| good | 2 | \( 1 + (48.0 + 83.2i)T + (-4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-45.1 + 78.1i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (2.83e4 + 4.90e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-4.96e4 + 8.59e4i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 + 2.66e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-2.70e7 + 4.67e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.99e7 + 3.45e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (4.74e8 + 8.21e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + 3.09e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-3.51e9 + 6.08e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-1.38e9 - 2.39e9i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 1.80e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 6.16e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-6.21e9 - 1.07e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-6.25e10 + 1.08e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-6.18e10 + 1.07e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (2.02e11 + 3.51e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (2.08e11 - 3.61e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 - 1.38e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-8.27e11 + 1.43e12i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-3.19e11 - 5.52e11i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 + 1.40e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.94e12 - 3.36e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 - 2.13e12T + 6.73e25T^{2} \) |
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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
−18.63653548502407251506132573862, −16.67745759317851852665234933744, −15.22280229271340168566797663432, −12.61201352205132463226866618212, −11.70769934747526130048868224255, −9.677640399007670355415467882009, −8.197415363712204464265067405739, −4.94538298644586638691496092966, −2.11327427001491387277366213171, −0.22459624757249313285022489012,
3.51208530566698915471609198408, 6.82830998858619157066304446564, 7.64468660293593088521301482910, 10.05238018538452528239665326946, 11.91876699918261207619134575764, 14.56068592292408680411293140924, 15.42808295853910794385705365910, 17.08244937136597327220503929032, 18.26705252222950242638618905481, 19.72304763574661967459070200148
