| L(s) = 1 | + (−10.7 − 11.3i)3-s − 4.25·5-s + 187. i·7-s + (−12.5 + 242. i)9-s + 712. i·11-s + 804. i·13-s + (45.6 + 48.0i)15-s − 309. i·17-s + 2.46e3·19-s + (2.11e3 − 2.01e3i)21-s − 3.06e3·23-s − 3.10e3·25-s + (2.87e3 − 2.46e3i)27-s − 6.32e3·29-s − 9.00e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.688 − 0.725i)3-s − 0.0760·5-s + 1.44i·7-s + (−0.0516 + 0.998i)9-s + 1.77i·11-s + 1.31i·13-s + (0.0523 + 0.0551i)15-s − 0.259i·17-s + 1.56·19-s + (1.04 − 0.995i)21-s − 1.20·23-s − 0.994·25-s + (0.759 − 0.650i)27-s − 1.39·29-s − 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6741293439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6741293439\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (10.7 + 11.3i)T \) |
| good | 5 | \( 1 + 4.25T + 3.12e3T^{2} \) |
| 7 | \( 1 - 187. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 712. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 804. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 309. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.00e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.35e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.03e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.10e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 2.07e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 9.40e4T + 8.58e9T^{2} \) |
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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
−11.58867715796273622668113562370, −9.792490074870850540459502984678, −9.395010370549941941894752003172, −7.934226418698487123576035718156, −7.24074122213567059705708257788, −6.16132259024044028122453509086, −5.35267978165273672622610326609, −4.28124891359274996513221220765, −2.33088564909798709594773085029, −1.71137927816167514707594846579,
0.21895160265744739638243331577, 0.977731987781142867207398262316, 3.40044770870606459406693805747, 3.82303961778962189139770629889, 5.35397439752172235565701666697, 5.92083261101689397327089319480, 7.27160282430748456355393641660, 8.134396579367021953412551411222, 9.361829783340041464117052897744, 10.35791738043395362651873571727
