| L(s) = 1 | + (1.72 + 5.38i)2-s + (−3.10 − 15.2i)3-s + (−26.0 + 18.5i)4-s + (−20.1 + 11.6i)5-s + (76.9 − 43.1i)6-s + (−156. − 90.5i)7-s + (−145. − 108. i)8-s + (−223. + 94.9i)9-s + (−97.4 − 88.4i)10-s + (41.2 − 71.4i)11-s + (364. + 340. i)12-s + (−70.1 − 121. i)13-s + (217. − 1.00e3i)14-s + (240. + 271. i)15-s + (332. − 968. i)16-s − 901. i·17-s + ⋯ |
| L(s) = 1 | + (0.305 + 0.952i)2-s + (−0.199 − 0.979i)3-s + (−0.813 + 0.581i)4-s + (−0.360 + 0.208i)5-s + (0.872 − 0.488i)6-s + (−1.20 − 0.698i)7-s + (−0.801 − 0.597i)8-s + (−0.920 + 0.390i)9-s + (−0.308 − 0.279i)10-s + (0.102 − 0.178i)11-s + (0.731 + 0.681i)12-s + (−0.115 − 0.199i)13-s + (0.296 − 1.36i)14-s + (0.275 + 0.311i)15-s + (0.324 − 0.945i)16-s − 0.756i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.103618 - 0.229895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103618 - 0.229895i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.72 - 5.38i)T \) |
| 3 | \( 1 + (3.10 + 15.2i)T \) |
| good | 5 | \( 1 + (20.1 - 11.6i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (156. + 90.5i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-41.2 + 71.4i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (70.1 + 121. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 901. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.36e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-160. - 277. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (7.03e3 + 4.06e3i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.53e3 + 885. i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.37e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (4.37e3 - 2.52e3i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.72e4 + 9.96e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-6.62e3 + 1.14e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 2.50e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (2.01e4 + 3.49e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (8.93e3 - 1.54e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.90e4 - 1.67e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.30e4 + 1.90e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.11e4 - 1.92e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 9.89e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (2.07e4 - 3.59e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−14.96351756955203212892994827174, −13.66753549210784174535880745076, −12.97252919005672922004068802346, −11.71166068785778307035273781411, −9.713340071463397071750747241787, −7.941086073020624923867033388443, −6.98599321958688030015490455650, −5.81810061292741725543789145816, −3.53504441506459247461914169898, −0.13217848030905343339526076811,
2.98555269040233507322961567653, 4.45005352201161364747350405648, 6.01748907008231180170727209570, 8.889278342260386048032666495599, 9.696270967186676133874792987623, 10.99528298444220324283250213659, 12.12371313800330810891749870919, 13.18709772433510324935816474067, 14.81850137578990468501195186428, 15.67103273978701123746990858637
