| L(s) = 1 | + (−1.10 − 0.877i)2-s + (0.272 + 3.11i)3-s + (0.459 + 1.94i)4-s + (−0.460 + 0.986i)5-s + (2.43 − 3.69i)6-s + (2.46 + 4.27i)7-s + (1.19 − 2.56i)8-s + (−6.68 + 1.17i)9-s + (1.37 − 0.690i)10-s + (0.785 − 0.210i)11-s + (−5.94 + 1.96i)12-s + (0.370 − 4.23i)13-s + (1.01 − 6.90i)14-s + (−3.20 − 1.16i)15-s + (−3.57 + 1.78i)16-s + (0.232 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.784 − 0.620i)2-s + (0.157 + 1.79i)3-s + (0.229 + 0.973i)4-s + (−0.205 + 0.441i)5-s + (0.993 − 1.50i)6-s + (0.932 + 1.61i)7-s + (0.424 − 0.905i)8-s + (−2.22 + 0.393i)9-s + (0.435 − 0.218i)10-s + (0.236 − 0.0634i)11-s + (−1.71 + 0.566i)12-s + (0.102 − 1.17i)13-s + (0.271 − 1.84i)14-s + (−0.826 − 0.300i)15-s + (−0.894 + 0.446i)16-s + (0.0564 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.435857 + 0.817621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435857 + 0.817621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 + 0.877i)T \) |
| 19 | \( 1 + (-4.33 + 0.467i)T \) |
| good | 3 | \( 1 + (-0.272 - 3.11i)T + (-2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (0.460 - 0.986i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-2.46 - 4.27i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.785 + 0.210i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.370 + 4.23i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.232 + 1.31i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (7.35 + 2.67i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.216i)T + (9.91 + 27.2i)T^{2} \) |
| 31 | \( 1 + (-1.07 - 1.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.57 + 1.57i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.34 - 4.48i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.0191 - 0.00894i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-9.28 + 1.63i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.61 + 5.60i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (0.695 + 0.993i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (3.49 + 7.49i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (0.0614 - 0.0877i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.212 - 0.585i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.62 - 4.32i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.61 - 6.38i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.78 + 2.35i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-5.35 + 4.49i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-13.0 - 2.29i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.60062565614074207397594253658, −10.99687332471038382915349414823, −10.14670976198251577058948996586, −9.326879225616602952937478482460, −8.589292805031113558817333627562, −7.83929543803006822205935562048, −5.80352732319493434670996959259, −4.80837240203190830905107311571, −3.45290655081561169230032190457, −2.55698656921964148092530234166,
0.930055618164587807874761656283, 1.82763704718235111243171178172, 4.32891738983858673209951256337, 5.89969518879022799408249065927, 6.93315248392019264429430481607, 7.57523140625626579456691698749, 8.112368026409197267763837903548, 9.140024544144735555069519612102, 10.48842337720526939507326311498, 11.52716982439393412687911407099
