| L(s) = 1 | + (−6.01 + 9.58i)2-s + (−22.2 + 53.8i)3-s + (−55.6 − 115. i)4-s + (416. − 172. i)5-s + (−381. − 537. i)6-s + (881. − 881. i)7-s + (1.43e3 + 160. i)8-s + (−853. − 853. i)9-s + (−851. + 5.02e3i)10-s + (−305. − 738. i)11-s + (7.44e3 − 424. i)12-s + (−3.35e3 − 1.38e3i)13-s + (3.14e3 + 1.37e4i)14-s + 2.62e4i·15-s + (−1.01e4 + 1.28e4i)16-s − 3.64e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.531 + 0.846i)2-s + (−0.476 + 1.15i)3-s + (−0.434 − 0.900i)4-s + (1.48 − 0.616i)5-s + (−0.721 − 1.01i)6-s + (0.971 − 0.971i)7-s + (0.993 + 0.110i)8-s + (−0.390 − 0.390i)9-s + (−0.269 + 1.58i)10-s + (−0.0692 − 0.167i)11-s + (1.24 − 0.0708i)12-s + (−0.423 − 0.175i)13-s + (0.306 + 1.33i)14-s + 2.00i·15-s + (−0.622 + 0.782i)16-s − 1.79i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.37144 + 0.686065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37144 + 0.686065i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.01 - 9.58i)T \) |
| good | 3 | \( 1 + (22.2 - 53.8i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-416. + 172. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-881. + 881. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (305. + 738. i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (3.35e3 + 1.38e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 3.64e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-4.18e4 - 1.73e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.47e4 - 4.47e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-4.88e3 + 1.17e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 3.91e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (6.37e4 - 2.63e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (2.14e5 + 2.14e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-2.85e5 - 6.88e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.48e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-6.10e5 - 1.47e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.32e5 + 5.50e4i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-1.17e6 + 2.83e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.10e6 + 2.67e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (2.02e6 - 2.02e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (2.16e5 + 2.16e5i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 3.34e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (3.39e6 + 1.40e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (6.24e6 - 6.24e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 2.74e6T + 8.07e13T^{2} \) |
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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
−15.82298639073490588868191628973, −14.26679366157898738262954504951, −13.60834543993730476601773227735, −11.13773600345971821219078742785, −9.943910373480004965603238695446, −9.344584735076462994517626782145, −7.45830650745432220649163431285, −5.43941284365637367354416736270, −4.84578750505372447348951765854, −1.12596015111755466098333311018,
1.45527766625993900666670640207, 2.35883946965283091529593350403, 5.51269790011341542878880818938, 7.05026206742287344292860008420, 8.676775493025661234181602524283, 10.11794372810850749942435004656, 11.39747831477841581512115687787, 12.48702909815053001311111787443, 13.46085396714077169114160262689, 14.73433532433813583103809902551
