| L(s) = 1 | + (−1.17 − 0.984i)3-s + (1.76 − 0.642i)5-s + (−1.03 − 1.78i)7-s + (−0.113 − 0.642i)9-s + (1.03 − 1.78i)11-s + (1.17 − 0.984i)13-s + (−2.70 − 0.984i)15-s + (0.0603 − 0.342i)17-s + (3.75 + 2.20i)19-s + (−0.549 + 3.11i)21-s + (−2.76 − 1.00i)23-s + (−1.12 + 0.943i)25-s + (−2.79 + 4.84i)27-s + (1.81 + 10.3i)29-s + (2.72 + 4.72i)31-s + ⋯ |
| L(s) = 1 | + (−0.677 − 0.568i)3-s + (0.789 − 0.287i)5-s + (−0.390 − 0.675i)7-s + (−0.0377 − 0.214i)9-s + (0.311 − 0.538i)11-s + (0.325 − 0.273i)13-s + (−0.698 − 0.254i)15-s + (0.0146 − 0.0829i)17-s + (0.862 + 0.506i)19-s + (−0.119 + 0.679i)21-s + (−0.576 − 0.209i)23-s + (−0.224 + 0.188i)25-s + (−0.538 + 0.932i)27-s + (0.337 + 1.91i)29-s + (0.489 + 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.797554 - 0.572334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.797554 - 0.572334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.75 - 2.20i)T \) |
| good | 3 | \( 1 + (1.17 + 0.984i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-1.76 + 0.642i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.03 + 1.78i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 1.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 0.984i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0603 + 0.342i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (2.76 + 1.00i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 10.3i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.72 - 4.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + (7.64 + 6.41i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.52 + 2.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.897 - 5.09i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (7.29 + 2.65i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0457 - 0.259i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.97 - 3.63i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.22 - 6.93i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.07 + 2.20i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (11.7 + 9.88i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.64 - 8.09i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.54 + 11.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.52 - 2.95i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.301 + 1.71i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.76550430061855846225427880301, −11.93301484519415995579904827727, −10.80695708443089404926386826863, −9.819636440293850515213709660852, −8.724774074603560093102360280369, −7.23481096736289519545060528754, −6.26549056709653924535841829113, −5.36597365863544060662209046487, −3.50122588065360034917209215979, −1.19095224497518122953183781289,
2.40889278577928883525086837825, 4.35606722321986693376324920422, 5.66902061407774838042289441339, 6.39409523823413928925466406807, 7.998230819253827292375867109429, 9.589723532852257538997236302456, 9.909986523555852467451196030518, 11.25986032898061658737417553645, 11.94634071781096483014573422677, 13.28321869732411562415336622459
