| L(s) = 1 | + (1.23 + 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (−32.5 + 45.4i)5-s + (29.1 + 21.1i)6-s − 77.9·7-s + (−51.7 − 37.6i)8-s + (25.0 − 77.0i)9-s + (−213. − 67.6i)10-s + (−147. − 454. i)11-s + (−44.4 + 136. i)12-s + (207. − 637. i)13-s + (−96.3 − 296. i)14-s + (3.33 + 503. i)15-s + (79.1 − 243. i)16-s + (1.15e3 + 836. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.582 + 0.812i)5-s + (0.330 + 0.239i)6-s − 0.601·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.673 − 0.214i)10-s + (−0.368 − 1.13i)11-s + (−0.0892 + 0.274i)12-s + (0.340 − 1.04i)13-s + (−0.131 − 0.404i)14-s + (0.00382 + 0.577i)15-s + (0.0772 − 0.237i)16-s + (0.965 + 0.701i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.385154813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385154813\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (32.5 - 45.4i)T \) |
| good | 7 | \( 1 + 77.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + (147. + 454. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-207. + 637. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.15e3 - 836. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-501. - 364. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (533. + 1.64e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.87e3 + 2.81e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.27e3 + 929. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.70e3 + 8.31e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.82e3 + 5.62e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.73e3 - 3.43e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-3.72e3 + 2.70e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-5.68e3 + 1.75e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (9.41e3 + 2.89e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.95e4 - 3.59e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.27e4 + 9.26e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.51e4 + 7.73e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (5.23e4 - 3.80e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-9.30e3 - 6.75e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (2.27e4 + 7.00e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.30e5 - 9.46e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.23451203185647439898940468312, −10.90151245914511580912885391668, −9.902832407025021209692703333872, −8.330407752662779198251586618482, −7.86062250661392573641066982258, −6.57804719036915087567727403978, −5.69873732427192015754015070760, −3.72058208875030918498988934663, −2.95004542321978902328080220235, −0.43443314503659234140075447237,
1.40016287822322499345955862081, 3.05597585220055801780815471100, 4.25090411334444399722638521178, 5.18960082668376823800725779043, 7.04429511686241192554838545957, 8.297471918968370783566231236865, 9.431422704382147303041518875296, 9.965149210729650671546158703675, 11.45939352692171863836412580075, 12.21984464585314392356956275024
