| L(s) = 1 | + (−1.14 + 0.825i)2-s + (0.0652 + 0.579i)3-s + (0.637 − 1.89i)4-s + (−2.49 − 1.20i)5-s + (−0.552 − 0.611i)6-s + (−0.0752 − 0.0600i)7-s + (0.832 + 2.70i)8-s + (2.59 − 0.591i)9-s + (3.85 − 0.678i)10-s + (−2.17 − 3.45i)11-s + (1.13 + 0.245i)12-s + (1.17 − 5.14i)13-s + (0.135 + 0.00680i)14-s + (0.532 − 1.52i)15-s + (−3.18 − 2.41i)16-s + (0.426 − 0.426i)17-s + ⋯ |
| L(s) = 1 | + (−0.812 + 0.583i)2-s + (0.0376 + 0.334i)3-s + (0.318 − 0.947i)4-s + (−1.11 − 0.536i)5-s + (−0.225 − 0.249i)6-s + (−0.0284 − 0.0226i)7-s + (0.294 + 0.955i)8-s + (0.864 − 0.197i)9-s + (1.21 − 0.214i)10-s + (−0.654 − 1.04i)11-s + (0.328 + 0.0708i)12-s + (0.325 − 1.42i)13-s + (0.0363 + 0.00181i)14-s + (0.137 − 0.392i)15-s + (−0.796 − 0.604i)16-s + (0.103 − 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.603794 - 0.236373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.603794 - 0.236373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.14 - 0.825i)T \) |
| 29 | \( 1 + (0.583 + 5.35i)T \) |
| good | 3 | \( 1 + (-0.0652 - 0.579i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (2.49 + 1.20i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 + (0.0752 + 0.0600i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (2.17 + 3.45i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 5.14i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.426 + 0.426i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.09 - 0.574i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (0.176 + 0.366i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-0.0855 - 0.244i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (3.40 + 2.13i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (4.88 + 4.88i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.49 - 4.26i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (-6.43 - 10.2i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (3.39 - 7.04i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + (-4.15 + 0.467i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (-6.00 + 1.37i)T + (60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.93 + 8.49i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.92 + 0.673i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (3.66 - 5.83i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (5.25 + 6.59i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-6.45 + 2.25i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.0586 - 0.520i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−11.88961786748796898328298496449, −10.86772099766081137768868415696, −10.10461201644140864565867734768, −8.993859699984042681072231024505, −7.983415297578954933769978168161, −7.53840840951549837575714185096, −5.94907380839885687336365791914, −4.88624276406346461891197545166, −3.38076217486027218150442888623, −0.72886830977011762645258277795,
1.79540066309459025911835505970, 3.42341723938672525410742813148, 4.57466917116975762730499520394, 7.00190619966045128381789063010, 7.24039751148260400257216096393, 8.310255660956724261461204036084, 9.534049686791907756689340797694, 10.38093395853533947824025755004, 11.41107142965037752022607995178, 12.04127638171926564633953655511
