| L(s) = 1 | + (−1.23 + 3.80i)2-s + (7.28 + 5.29i)3-s + (−12.9 − 9.40i)4-s + (51.5 − 21.7i)5-s + (−29.1 + 21.1i)6-s − 38.5·7-s + (51.7 − 37.6i)8-s + (25.0 + 77.0i)9-s + (18.9 + 222. i)10-s + (100. − 308. i)11-s + (−44.4 − 136. i)12-s + (52.3 + 161. i)13-s + (47.6 − 146. i)14-s + (489. + 114. i)15-s + (79.1 + 243. i)16-s + (620. − 450. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.921 − 0.388i)5-s + (−0.330 + 0.239i)6-s − 0.297·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.0597 + 0.704i)10-s + (0.249 − 0.769i)11-s + (−0.0892 − 0.274i)12-s + (0.0859 + 0.264i)13-s + (0.0650 − 0.200i)14-s + (0.562 + 0.131i)15-s + (0.0772 + 0.237i)16-s + (0.520 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.324289066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.324289066\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 3 | \( 1 + (-7.28 - 5.29i)T \) |
| 5 | \( 1 + (-51.5 + 21.7i)T \) |
| good | 7 | \( 1 + 38.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-100. + 308. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-52.3 - 161. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-620. + 450. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-591. + 429. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-991. + 3.05e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.56e3 - 1.13e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.39e3 + 1.73e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.96e3 - 6.05e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.19e3 - 3.67e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.31e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.27e4 - 9.28e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.22e3 + 1.61e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-9.93e3 - 3.05e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (6.57e3 - 2.02e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.85e4 + 1.34e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.74e4 - 1.99e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.47e4 + 7.61e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.29e4 - 9.44e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.59e3 + 4.78e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (5.30e3 - 1.63e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.25e5 + 9.10e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.40178052612720187966750496045, −10.86556361145492097746970068868, −9.784700334510301304717164581475, −9.075731694567825897962456261372, −8.176997331468307723259769031350, −6.73357125879825695083251911735, −5.71823008641342322386219570613, −4.51484239075776085315389242388, −2.81967288982949377322016685236, −0.965181710134280595266698992186,
1.25891387047445230218023739261, 2.44073014551767974247792725511, 3.66499782699928777023732147568, 5.43929114551216753854706428477, 6.78815474546173183543797650890, 7.918463018258171110489175521882, 9.297557295784721154228508043407, 9.848913548982176143688254190341, 10.89625371106602544891408862480, 12.18474452447144478175796280234
