| L(s) = 1 | + (−36.4 + 63.2i)2-s + (−167. − 290. i)3-s + (−1.63e3 − 2.83e3i)4-s + (−539. + 933. i)5-s + 2.44e4·6-s + (4.44e4 − 388. i)7-s + 8.97e4·8-s + (3.22e4 − 5.59e4i)9-s + (−3.93e4 − 6.81e4i)10-s + (−4.13e5 − 7.16e5i)11-s + (−5.49e5 + 9.52e5i)12-s + 1.32e6·13-s + (−1.59e6 + 2.82e6i)14-s + 3.61e5·15-s + (8.04e4 − 1.39e5i)16-s + (−4.99e6 − 8.64e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.806 + 1.39i)2-s + (−0.398 − 0.690i)3-s + (−0.800 − 1.38i)4-s + (−0.0771 + 0.133i)5-s + 1.28·6-s + (0.999 − 0.00874i)7-s + 0.968·8-s + (0.182 − 0.315i)9-s + (−0.124 − 0.215i)10-s + (−0.774 − 1.34i)11-s + (−0.637 + 1.10i)12-s + 0.987·13-s + (−0.794 + 1.40i)14-s + 0.123·15-s + (0.0191 − 0.0332i)16-s + (−0.852 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.723667 - 0.166193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.723667 - 0.166193i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-4.44e4 + 388. i)T \) |
| good | 2 | \( 1 + (36.4 - 63.2i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (167. + 290. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (539. - 933. i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (4.13e5 + 7.16e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.32e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (4.99e6 + 8.64e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.37e6 - 4.10e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.93e6 + 3.35e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.61e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (8.44e7 + 1.46e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.41e8 + 4.18e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 3.92e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 3.00e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-9.27e8 + 1.60e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.14e9 - 1.98e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.76e9 - 3.05e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (3.52e9 - 6.10e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (3.55e9 + 6.15e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.47e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.17e9 + 2.04e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (4.25e8 - 7.36e8i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.73e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (2.25e10 - 3.91e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 5.64e10T + 7.15e21T^{2} \) |
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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
−18.48885796023327465928891186250, −18.21193478547455927606069198428, −16.65505250827764210999091146606, −15.33820078745846771142814898062, −13.65769062830342055333376508898, −11.23586944080326985463404064546, −8.731231242216262264368692274386, −7.31988075270399702937133469423, −5.74336420648257024057864017999, −0.64660509451440513376539552320,
1.81021971770840660388636620324, 4.46952584187828162304636625919, 8.401237243756517061589985953177, 10.27383750611384759822723476679, 11.11919517961835629108714038513, 12.81713610938873135452404255078, 15.39438246793135826167557432354, 17.32996351375191785767445243200, 18.33753748003139709161598877018, 20.02304388994889981347782871725
