| L(s) = 1 | + (0.975 − 1.02i)2-s + (−1.23 + 0.573i)3-s + (−0.0958 − 1.99i)4-s + (2.13 − 3.05i)5-s + (−0.613 + 1.81i)6-s + (−0.144 + 0.250i)7-s + (−2.13 − 1.85i)8-s + (−0.743 + 0.886i)9-s + (−1.04 − 5.17i)10-s + (2.30 + 0.617i)11-s + (1.26 + 2.40i)12-s + (−5.70 − 2.65i)13-s + (0.115 + 0.391i)14-s + (−0.879 + 4.98i)15-s + (−3.98 + 0.382i)16-s + (4.81 − 4.03i)17-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.723i)2-s + (−0.710 + 0.331i)3-s + (−0.0479 − 0.998i)4-s + (0.956 − 1.36i)5-s + (−0.250 + 0.742i)6-s + (−0.0545 + 0.0945i)7-s + (−0.756 − 0.654i)8-s + (−0.247 + 0.295i)9-s + (−0.328 − 1.63i)10-s + (0.695 + 0.186i)11-s + (0.364 + 0.693i)12-s + (−1.58 − 0.737i)13-s + (0.0307 + 0.104i)14-s + (−0.227 + 1.28i)15-s + (−0.995 + 0.0957i)16-s + (1.16 − 0.979i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.877289 - 1.29665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.877289 - 1.29665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.975 + 1.02i)T \) |
| 19 | \( 1 + (-4.34 + 0.395i)T \) |
| good | 3 | \( 1 + (1.23 - 0.573i)T + (1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (-2.13 + 3.05i)T + (-1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (0.144 - 0.250i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 0.617i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (5.70 + 2.65i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-4.81 + 4.03i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.479 - 2.71i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.329 - 3.77i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (3.76 - 6.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.56 + 4.56i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.45 - 1.98i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-4.84 - 3.39i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (1.11 - 1.32i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.56 - 5.09i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-1.05 - 0.0919i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.26 - 4.66i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-5.46 + 0.478i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (11.0 - 1.94i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.145 + 0.399i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (14.0 + 5.12i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-14.2 + 3.82i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.679 + 0.247i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.86 - 3.41i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.75152573154655861404087535180, −10.49801337144177580675533714067, −9.641362160387376903245318461145, −9.197844389605272047269254952438, −7.45201940246408387533577119368, −5.71367820169547080018808423405, −5.34685243936502309653791458453, −4.58489697909422449857760801396, −2.75248475944621143385572515548, −1.09149441054176667778532461892,
2.46660356843053324336990170592, 3.77681896826711554190343197375, 5.43704565622275247644941244140, 6.13944211220953530921874764593, 6.85412357838280602974694206625, 7.65172985951171433447306240747, 9.303533173605577120739961386489, 10.13473062013066121744464776201, 11.44500286635611081287106890222, 11.96936880295010365082861349362
