| L(s) = 1 | + (−0.642 + 1.26i)2-s + (−0.270 + 1.71i)3-s + (−1.17 − 1.61i)4-s + (4.98 + 0.362i)5-s + (−1.98 − 1.43i)6-s + (4.91 + 4.91i)7-s + (2.79 − 0.442i)8-s + (−2.85 − 0.927i)9-s + (−3.65 + 6.05i)10-s + (−0.450 − 1.38i)11-s + (3.08 − 1.57i)12-s + (4.46 + 8.77i)13-s + (−9.35 + 3.04i)14-s + (−1.97 + 8.43i)15-s + (−1.23 + 3.80i)16-s + (0.887 + 5.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (−0.0903 + 0.570i)3-s + (−0.293 − 0.404i)4-s + (0.997 + 0.0724i)5-s + (−0.330 − 0.239i)6-s + (0.702 + 0.702i)7-s + (0.349 − 0.0553i)8-s + (−0.317 − 0.103i)9-s + (−0.365 + 0.605i)10-s + (−0.0409 − 0.125i)11-s + (0.257 − 0.131i)12-s + (0.343 + 0.674i)13-s + (−0.668 + 0.217i)14-s + (−0.131 + 0.562i)15-s + (−0.0772 + 0.237i)16-s + (0.0521 + 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.859699 + 1.08643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.859699 + 1.08643i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 1.26i)T \) |
| 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (-4.98 - 0.362i)T \) |
| good | 7 | \( 1 + (-4.91 - 4.91i)T + 49iT^{2} \) |
| 11 | \( 1 + (0.450 + 1.38i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-4.46 - 8.77i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-0.887 - 5.60i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (15.9 - 21.9i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (20.8 + 10.6i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (16.3 + 22.5i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-24.0 - 17.4i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-64.2 + 32.7i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.47 + 4.52i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (4.91 - 4.91i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-49.7 - 7.88i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-15.2 + 96.2i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (53.3 + 17.3i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-7.49 - 23.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-7.62 - 48.1i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-45.6 + 33.1i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (108. + 55.5i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (69.8 + 96.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-79.7 + 12.6i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (17.8 - 5.79i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-1.77 - 0.281i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−13.29264254130229770720058646051, −11.98058383952061453095200771346, −10.78157585453166454677461762136, −9.893326753771666587889967776963, −8.919836672357954370056445066710, −8.064235599048928881561808587472, −6.31601617368280862636541964399, −5.65544154461910043716317221441, −4.30105143833871986513710259219, −2.02554900978655228637992551731,
1.13050666872105729572226749943, 2.57325051733855327951037612489, 4.53242987612279628693441849893, 5.94828520786071762821367147176, 7.31265884293056218476620020775, 8.375795614552310294500146298180, 9.529153102035011140426488117148, 10.58606007697194483309845773202, 11.31734587917027655038807017689, 12.57749345320915644606684525613
