| L(s) = 1 | + (58.8 + 55.6i)3-s + (−726. + 419. i)5-s + (−1.18e3 + 2.08e3i)7-s + (356. + 6.55e3i)9-s + (1.18e4 + 6.84e3i)11-s − 4.28e3·13-s + (−6.60e4 − 1.57e4i)15-s + (−2.08e4 − 1.20e4i)17-s + (−6.64e4 − 1.15e5i)19-s + (−1.86e5 + 5.65e4i)21-s + (−3.13e4 + 1.80e4i)23-s + (1.56e5 − 2.70e5i)25-s + (−3.43e5 + 4.05e5i)27-s − 1.12e6i·29-s + (−1.48e5 + 2.57e5i)31-s + ⋯ |
| L(s) = 1 | + (0.726 + 0.687i)3-s + (−1.16 + 0.670i)5-s + (−0.494 + 0.868i)7-s + (0.0543 + 0.998i)9-s + (0.809 + 0.467i)11-s − 0.150·13-s + (−1.30 − 0.311i)15-s + (−0.249 − 0.143i)17-s + (−0.510 − 0.883i)19-s + (−0.956 + 0.290i)21-s + (−0.111 + 0.0646i)23-s + (0.399 − 0.692i)25-s + (−0.647 + 0.762i)27-s − 1.59i·29-s + (−0.160 + 0.278i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.250715 - 0.632237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250715 - 0.632237i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-58.8 - 55.6i)T \) |
| 7 | \( 1 + (1.18e3 - 2.08e3i)T \) |
| good | 5 | \( 1 + (726. - 419. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.18e4 - 6.84e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.28e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + (2.08e4 + 1.20e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (6.64e4 + 1.15e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.13e4 - 1.80e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.12e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (1.48e5 - 2.57e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.66e6 - 2.87e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.70e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.75e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (2.29e6 - 1.32e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.36e6 + 2.51e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.72e7 + 9.93e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.29e6 - 1.08e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.30e7 - 2.26e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.53e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.43e7 + 4.21e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.10e7 - 3.64e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.42e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-3.35e7 + 1.93e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 9.15e7T + 7.83e15T^{2} \) |
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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
−13.40128199332224335806760609953, −12.03205990057398545084137773577, −11.18761968162414591701634775493, −9.861853462263467537036114041901, −8.900111993569445895037984596626, −7.78702125725773471426198524389, −6.53101136526933982566815500865, −4.63292990703851855444300449317, −3.52260359300649444889007007075, −2.39919751669677526193193359710,
0.19119439601033112256630169799, 1.35990809295992101850671381974, 3.38346487028877082287311949203, 4.23798492370884673420136244936, 6.39015992693150184491266311577, 7.51803535620883992923848076784, 8.393375456738831289931785785167, 9.442682861708055436574873179212, 11.03578252144755442824212876198, 12.25353831651511088177056790907
