| L(s) = 1 | + (3.05 + 2.57i)2-s + (2.72 + 15.7i)4-s + (−14.3 + 24.7i)5-s + (22.2 − 12.8i)7-s + (−32.3 + 55.2i)8-s + (−107. + 38.9i)10-s + (−93.9 + 54.2i)11-s + (44.2 − 76.5i)13-s + (101. + 17.9i)14-s + (−241. + 85.8i)16-s − 504.·17-s + 191. i·19-s + (−429. − 158. i)20-s + (−427. − 76.1i)22-s + (831. + 480. i)23-s + ⋯ |
| L(s) = 1 | + (0.764 + 0.644i)2-s + (0.170 + 0.985i)4-s + (−0.572 + 0.991i)5-s + (0.453 − 0.261i)7-s + (−0.504 + 0.863i)8-s + (−1.07 + 0.389i)10-s + (−0.776 + 0.448i)11-s + (0.261 − 0.453i)13-s + (0.515 + 0.0918i)14-s + (−0.942 + 0.335i)16-s − 1.74·17-s + 0.530i·19-s + (−1.07 − 0.395i)20-s + (−0.882 − 0.157i)22-s + (1.57 + 0.907i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.348227 + 1.87938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.348227 + 1.87938i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.05 - 2.57i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (14.3 - 24.7i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-22.2 + 12.8i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (93.9 - 54.2i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-44.2 + 76.5i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 504.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 191. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-831. - 480. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-396. - 687. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (285. + 164. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 209.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-528. + 914. i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-2.88e3 + 1.66e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-977. + 564. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.13e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.03e3 - 2.33e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.79e3 - 4.84e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-6.12e3 - 3.53e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.95e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (1.52e3 - 879. i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (2.62e3 - 1.51e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 559.T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.10e3 - 1.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.52422124295123544839435121662, −12.66416167122806011578200541622, −11.28853940207651931277128480272, −10.72839806190527451696355288332, −8.789025963538064972841674084327, −7.52883291767212073974997069600, −6.90697896588268558947198241150, −5.37065342996470640757533284895, −4.04895547609738533655711272473, −2.68809270268556389292393620842,
0.65859066781221928549747609964, 2.49279553383707931645145450583, 4.32610788062708071370311770569, 5.06868609070256536020978165305, 6.60788093935198423327043695747, 8.363325964487692113425673874478, 9.243806598389651254007357554890, 10.92281455808950614095220855338, 11.42346481220820113230188175165, 12.72574874684154893569735902433
