| L(s) = 1 | + (4 + 6.92i)2-s + (35.5 + 61.6i)3-s + (−31.9 + 55.4i)4-s + 523.·5-s + (−284. + 492. i)6-s + (208. − 360. i)7-s − 511.·8-s + (−1.43e3 + 2.49e3i)9-s + (2.09e3 + 3.62e3i)10-s + (−1.71e3 − 2.96e3i)11-s − 4.55e3·12-s + (−6.74e3 − 4.15e3i)13-s + 3.33e3·14-s + (1.86e4 + 3.22e4i)15-s + (−2.04e3 − 3.54e3i)16-s + (3.26e3 − 5.66e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.760 + 1.31i)3-s + (−0.249 + 0.433i)4-s + 1.87·5-s + (−0.537 + 0.931i)6-s + (0.229 − 0.397i)7-s − 0.353·8-s + (−0.657 + 1.13i)9-s + (0.662 + 1.14i)10-s + (−0.387 − 0.671i)11-s − 0.760·12-s + (−0.851 − 0.525i)13-s + 0.324·14-s + (1.42 + 2.46i)15-s + (−0.125 − 0.216i)16-s + (0.161 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.86555 + 2.37452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86555 + 2.37452i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 13 | \( 1 + (6.74e3 + 4.15e3i)T \) |
| good | 3 | \( 1 + (-35.5 - 61.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 523.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-208. + 360. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.71e3 + 2.96e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-3.26e3 + 5.66e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.39e4 - 2.42e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.30e4 + 9.19e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.42e4 + 5.93e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 5.15e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.40e4 + 4.16e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-3.01e5 - 5.21e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.58e5 - 7.93e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 3.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.34e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (5.87e5 - 1.01e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.44e6 + 2.50e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.59e5 + 2.75e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (6.41e5 - 1.11e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 1.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.93e6 + 6.82e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-3.40e6 + 5.89e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−16.25029011579572675279617308888, −14.65132698796591748108293531572, −14.21201292946650620119916019944, −13.01596632490984819616057486234, −10.40711909285565494612530268500, −9.695427781029161976108185641649, −8.311122336715776955996242379328, −6.01761739580206786286193250518, −4.67975183157922891483601731651, −2.71018029187727279853891285228,
1.74495065676387056838552167449, 2.38291324254866173764632379237, 5.46484886820953479772836985539, 7.00867061791101382846931839147, 8.942488827783357871327831757949, 10.08497043803827190675439014939, 12.12843413966603210069818228028, 13.18770264774338598207825048793, 13.83470799088299567448309680381, 14.84676666327333063445700580191
