| L(s) = 1 | + (−1.28 + 0.592i)2-s + (0.987 + 0.156i)3-s + (1.29 − 1.52i)4-s + (−1.17 − 1.89i)5-s + (−1.36 + 0.384i)6-s + (1.39 − 1.39i)7-s + (−0.763 + 2.72i)8-s + (0.951 + 0.309i)9-s + (2.64 + 1.73i)10-s + (−1.17 + 0.382i)11-s + (1.51 − 1.30i)12-s + (−2.63 − 5.16i)13-s + (−0.966 + 2.62i)14-s + (−0.867 − 2.06i)15-s + (−0.633 − 3.94i)16-s + (−0.916 − 5.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)2-s + (0.570 + 0.0903i)3-s + (0.648 − 0.761i)4-s + (−0.527 − 0.849i)5-s + (−0.555 + 0.157i)6-s + (0.528 − 0.528i)7-s + (−0.269 + 0.962i)8-s + (0.317 + 0.103i)9-s + (0.835 + 0.550i)10-s + (−0.355 + 0.115i)11-s + (0.438 − 0.375i)12-s + (−0.729 − 1.43i)13-s + (−0.258 + 0.700i)14-s + (−0.224 − 0.532i)15-s + (−0.158 − 0.987i)16-s + (−0.222 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.817203 - 0.417829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817203 - 0.417829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.592i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (1.17 + 1.89i)T \) |
| good | 7 | \( 1 + (-1.39 + 1.39i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.17 - 0.382i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.63 + 5.16i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.916 + 5.78i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-3.68 - 2.68i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.0552 + 0.108i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-3.15 - 4.34i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.30 + 8.67i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.90 - 1.98i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 5.81i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.87 - 6.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.721 + 4.55i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.00324 + 0.0205i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (2.84 - 8.75i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.217 - 0.668i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 1.11i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-4.63 - 6.38i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.73 + 2.41i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (10.1 - 7.39i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.455 - 2.87i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-9.16 + 2.97i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (7.28 + 1.15i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.48293253735635666741805063055, −10.35813351149066462057899117617, −9.626979112517828996320318849614, −8.627594310636297848587854126361, −7.70425968361274402404283541093, −7.40777721130398781697427300678, −5.53700574402256280703276183406, −4.62927162260548644238902029633, −2.79412133201750436574658378505, −0.882405399571699608234620133594,
1.98623027566906298855703768122, 3.05977183644004241148581637181, 4.40204513011220895919138525564, 6.43173983965317562614272789754, 7.30902555756837412067878204097, 8.195954538005081792381819154087, 8.968355449847994241459635158720, 10.01596504191649564162512022596, 10.88674819744296548148247044455, 11.77284659061626116902060146129
