| L(s) = 1 | + (0.818 + 1.15i)2-s + (1.61 + 0.751i)3-s + (−0.661 + 1.88i)4-s + (−0.488 − 0.697i)5-s + (0.451 + 2.47i)6-s + (1.27 + 2.21i)7-s + (−2.71 + 0.781i)8-s + (0.104 + 0.124i)9-s + (0.405 − 1.13i)10-s + (0.0771 − 0.0206i)11-s + (−2.48 + 2.54i)12-s + (1.07 − 0.500i)13-s + (−1.50 + 3.28i)14-s + (−0.262 − 1.49i)15-s + (−3.12 − 2.49i)16-s + (−1.69 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (0.578 + 0.815i)2-s + (0.930 + 0.433i)3-s + (−0.330 + 0.943i)4-s + (−0.218 − 0.311i)5-s + (0.184 + 1.01i)6-s + (0.482 + 0.836i)7-s + (−0.961 + 0.276i)8-s + (0.0348 + 0.0415i)9-s + (0.128 − 0.358i)10-s + (0.0232 − 0.00623i)11-s + (−0.717 + 0.734i)12-s + (0.297 − 0.138i)13-s + (−0.402 + 0.877i)14-s + (−0.0679 − 0.385i)15-s + (−0.781 − 0.624i)16-s + (−0.410 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36322 + 1.63867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36322 + 1.63867i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.818 - 1.15i)T \) |
| 19 | \( 1 + (-4.17 + 1.24i)T \) |
| good | 3 | \( 1 + (-1.61 - 0.751i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (0.488 + 0.697i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.27 - 2.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0771 + 0.0206i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 0.500i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (1.69 + 1.42i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.869 - 4.93i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.112 - 1.28i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (2.07 + 3.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.759 - 0.759i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.07 - 0.390i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.60 + 2.52i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (5.67 + 6.76i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (6.28 - 8.97i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-9.11 + 0.797i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (4.34 - 6.20i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.275 + 0.0240i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.26 - 0.751i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 5.45i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.86 + 2.13i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (13.1 + 3.53i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (13.7 + 4.99i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.531 - 0.632i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−12.04738336429264361507683490260, −11.39396783011882864108717012007, −9.645509645163103503795224663429, −8.867698828938481125007622861636, −8.278192947313792008355660988158, −7.27570362100014499691569935092, −5.89056204369167358537628193780, −4.93439754091613318559646510750, −3.77114198134015445059912364686, −2.66653264452721247320104335826,
1.51231951063412243066342605756, 2.89667392048965018472598734895, 3.90337956177588706687035600001, 5.11580091343137200835951745828, 6.61322124400390531651061209276, 7.68108367935139058278535253749, 8.655528500660210384335699846848, 9.689313770901142613074673994018, 10.84876796156597952056714376208, 11.28250357160032292551924045819
