| L(s) = 1 | + 11.1i·3-s + (−5 + 55.6i)5-s + 122. i·7-s + 119.·9-s − 100·11-s − 734. i·13-s + (−619. − 55.6i)15-s − 979. i·17-s + 2.24e3·19-s − 1.36e3·21-s + 3.41e3i·23-s + (−3.07e3 − 556. i)25-s + 4.03e3i·27-s + 7.85e3·29-s − 2.14e3·31-s + ⋯ |
| L(s) = 1 | + 0.714i·3-s + (−0.0894 + 0.995i)5-s + 0.944i·7-s + 0.489·9-s − 0.249·11-s − 1.20i·13-s + (−0.711 − 0.0638i)15-s − 0.822i·17-s + 1.42·19-s − 0.674·21-s + 1.34i·23-s + (−0.983 − 0.178i)25-s + 1.06i·27-s + 1.73·29-s − 0.400·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.00510 + 0.918892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00510 + 0.918892i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (5 - 55.6i)T \) |
| good | 3 | \( 1 - 11.1iT - 243T^{2} \) |
| 7 | \( 1 - 122. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 100T + 1.61e5T^{2} \) |
| 13 | \( 1 + 734. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 979. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.04e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.43e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.42e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.62e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 826T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.75e4iT - 8.58e9T^{2} \) |
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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
−17.84363522348300228951559373205, −15.74521455781585049498192770458, −15.42233875909483698139822429855, −13.87188415180304364461113129683, −12.08874978997987416003300584248, −10.64504106758574878652752842428, −9.481164855293540978157935685805, −7.47564668307100045902375505283, −5.42939889210902408765268137806, −3.12267090635232588918343232799,
1.20885337565256542323483111060, 4.47437149197485388780857901874, 6.79186572701970879671106997014, 8.247989712868395510428902988242, 10.00149650160282765319608589309, 11.87786362481632832740887515917, 13.03291037074093124952172234397, 14.03109062416485832697968411262, 15.99604949966123306924788322643, 16.93054844274819943226095783149
