| L(s) = 1 | + (−24.3 − 42.2i)2-s + (1.09e3 − 1.90e3i)3-s + (2.90e3 − 5.03e3i)4-s + (1.29e4 + 2.25e4i)5-s − 1.07e5·6-s + (2.82e5 − 1.30e5i)7-s − 6.82e5·8-s + (−1.61e6 − 2.80e6i)9-s + (6.33e5 − 1.09e6i)10-s + (−3.13e6 + 5.43e6i)11-s + (−6.39e6 − 1.10e7i)12-s + 6.36e6·13-s + (−1.23e7 − 8.75e6i)14-s + 5.71e7·15-s + (−7.18e6 − 1.24e7i)16-s + (−3.56e7 + 6.17e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.269 − 0.466i)2-s + (0.870 − 1.50i)3-s + (0.354 − 0.614i)4-s + (0.371 + 0.644i)5-s − 0.937·6-s + (0.907 − 0.419i)7-s − 0.920·8-s + (−1.01 − 1.75i)9-s + (0.200 − 0.346i)10-s + (−0.533 + 0.924i)11-s + (−0.618 − 1.07i)12-s + 0.365·13-s + (−0.439 − 0.310i)14-s + 1.29·15-s + (−0.107 − 0.185i)16-s + (−0.358 + 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.931389 - 1.94625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.931389 - 1.94625i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.82e5 + 1.30e5i)T \) |
| good | 2 | \( 1 + (24.3 + 42.2i)T + (-4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-1.09e3 + 1.90e3i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.29e4 - 2.25e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (3.13e6 - 5.43e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 - 6.36e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (3.56e7 - 6.17e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-1.61e8 - 2.79e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (3.37e8 + 5.85e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 - 4.62e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-1.77e9 + 3.08e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-3.51e9 - 6.08e9i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 - 9.68e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.40e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.76e10 + 3.04e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (6.27e10 - 1.08e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (1.04e11 - 1.80e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (4.14e10 + 7.17e10i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (4.25e11 - 7.37e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 - 4.45e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-1.02e12 + 1.77e12i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-1.37e12 - 2.37e12i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 + 2.03e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.63e12 - 2.83e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + 4.35e11T + 6.73e25T^{2} \) |
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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
−18.50292862907718119759257560144, −17.98005034691452118084497281104, −14.85409568047948658728426210971, −13.91640868938940583121929252646, −12.13336190733142846640131035324, −10.30387691169818460207065227664, −8.063239401536108949947193207457, −6.49761664203356067024847592735, −2.42674006926585281262939506240, −1.32761355097120706483926348613,
2.95599967070192070348430192422, 5.04304967146406007229729435085, 8.244781330738273477387840017745, 9.176702785552520094579656498602, 11.26922410459297482329849406461, 13.78180909578132737900120102818, 15.49336428364918400347512507463, 16.18581178822347856615032975250, 17.73116012443957900830123362904, 20.14141600176223105364680173664
