| L(s) = 1 | + (0.909 − 0.415i)2-s + (0.642 + 2.18i)3-s + (0.654 − 0.755i)4-s + (0.0816 − 2.23i)5-s + (1.49 + 1.72i)6-s + (2.70 − 0.388i)7-s + (0.281 − 0.959i)8-s + (−1.85 + 1.19i)9-s + (−0.854 − 2.06i)10-s + (−1.73 + 3.80i)11-s + (2.07 + 0.947i)12-s + (−4.03 − 0.579i)13-s + (2.29 − 1.47i)14-s + (4.94 − 1.25i)15-s + (−0.142 − 0.989i)16-s + (1.45 − 1.25i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.293i)2-s + (0.371 + 1.26i)3-s + (0.327 − 0.377i)4-s + (0.0365 − 0.999i)5-s + (0.609 + 0.703i)6-s + (1.02 − 0.146i)7-s + (0.0996 − 0.339i)8-s + (−0.618 + 0.397i)9-s + (−0.270 − 0.653i)10-s + (−0.524 + 1.14i)11-s + (0.599 + 0.273i)12-s + (−1.11 − 0.160i)13-s + (0.614 − 0.394i)14-s + (1.27 − 0.324i)15-s + (−0.0355 − 0.247i)16-s + (0.351 − 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.97365 + 0.179772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97365 + 0.179772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (-0.0816 + 2.23i)T \) |
| 23 | \( 1 + (-4.14 + 2.41i)T \) |
| good | 3 | \( 1 + (-0.642 - 2.18i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-2.70 + 0.388i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.73 - 3.80i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (4.03 + 0.579i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 1.25i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.27 - 3.77i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.14 + 2.46i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (6.90 + 2.02i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.263 + 0.410i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.39 + 1.54i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.64 - 5.59i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 12.6iT - 47T^{2} \) |
| 53 | \( 1 + (0.0981 - 0.0141i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 9.49i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-4.44 - 1.30i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (10.0 - 4.57i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-5.81 - 12.7i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-9.10 - 7.88i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.565 + 3.93i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.85 - 9.10i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (4.46 - 1.31i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.30 + 6.69i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−12.36944458342670650645058963769, −11.26108264744948701254997541290, −10.17727596740947104058201454744, −9.626946590763685629183988906025, −8.451942707517150063872947482723, −7.35867736538039635493142224697, −5.28623087030906466981207712270, −4.80950180727547284550208526900, −3.93579287951344203166292935755, −2.11595749986336419119284531115,
2.05548534532745899410269725774, 3.15408270460525583347875420590, 5.00294236723702379619179813778, 6.19648885791936254670910440294, 7.27640052006723492986035059504, 7.74372962757258306497738886760, 8.873137970883370584594151451914, 10.69065115332594544862143235633, 11.33977603112004135088865510759, 12.38166432277696950902978834324
