| L(s) = 1 | + (−6.98 + 12.1i)3-s + (859. + 1.48e3i)5-s + (−1.80e3 + 6.09e3i)7-s + (9.74e3 + 1.68e4i)9-s + (−3.56e4 + 6.17e4i)11-s + 1.56e5·13-s − 2.40e4·15-s + (−2.55e5 + 4.43e5i)17-s + (−9.72e4 − 1.68e5i)19-s + (−6.11e4 − 6.43e4i)21-s + (−5.40e4 − 9.36e4i)23-s + (−5.00e5 + 8.67e5i)25-s − 5.47e5·27-s + 4.21e6·29-s + (1.58e6 − 2.74e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.0498 + 0.0862i)3-s + (0.614 + 1.06i)5-s + (−0.283 + 0.958i)7-s + (0.495 + 0.857i)9-s + (−0.734 + 1.27i)11-s + 1.52·13-s − 0.122·15-s + (−0.742 + 1.28i)17-s + (−0.171 − 0.296i)19-s + (−0.0685 − 0.0722i)21-s + (−0.0402 − 0.0697i)23-s + (−0.256 + 0.444i)25-s − 0.198·27-s + 1.10·29-s + (0.307 − 0.533i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.343i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.103787792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103787792\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.80e3 - 6.09e3i)T \) |
| good | 3 | \( 1 + (6.98 - 12.1i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-859. - 1.48e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.56e4 - 6.17e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.56e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (2.55e5 - 4.43e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (9.72e4 + 1.68e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (5.40e4 + 9.36e4i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-1.58e6 + 2.74e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (7.20e6 + 1.24e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 7.69e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.64e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-9.11e6 - 1.57e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.26e7 - 3.91e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (4.89e7 - 8.47e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (5.73e7 + 9.93e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.00e8 + 1.74e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.08e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (4.42e6 - 7.66e6i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.92e7 - 3.32e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-8.24e7 - 1.42e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.57e8T + 7.60e17T^{2} \) |
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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
−12.49044409239028285333655202989, −10.83994066649532980014858590489, −10.48424087996433123652346774512, −9.219394695947302422073055594937, −7.961915961154256766470031004149, −6.64406254550849312103340504544, −5.79269611499724634209634579632, −4.31671717641658742217640705166, −2.63918069325227259174741978121, −1.85138989928159099254302703264,
0.58326112921567431512309324550, 1.19577800606378995487195890702, 3.19676563716586737118920754537, 4.45768466992005862685242110796, 5.79062986402482790012473331673, 6.79470744711159924069072407572, 8.336228200101980711050919954834, 9.153103173796500080554751650845, 10.29462864939578051445425947769, 11.32395621046663692183680043141
