L(s) = 1 | + (1.29 − 1.15i)3-s + (1.99 − 1.00i)5-s + (0.814 − 0.814i)7-s + (0.345 − 2.98i)9-s + (−3.46 − 1.12i)11-s + (−2.63 + 5.16i)13-s + (1.42 − 3.60i)15-s + (0.902 + 0.142i)17-s + (4.29 + 5.91i)19-s + (0.115 − 1.99i)21-s + (−3.05 − 6.00i)23-s + (2.97 − 4.02i)25-s + (−2.98 − 4.25i)27-s + (4.30 + 3.12i)29-s + (−1.78 + 1.29i)31-s + ⋯ |
L(s) = 1 | + (0.746 − 0.665i)3-s + (0.892 − 0.450i)5-s + (0.307 − 0.307i)7-s + (0.115 − 0.993i)9-s + (−1.04 − 0.339i)11-s + (−0.730 + 1.43i)13-s + (0.367 − 0.930i)15-s + (0.218 + 0.0346i)17-s + (0.986 + 1.35i)19-s + (0.0251 − 0.434i)21-s + (−0.637 − 1.25i)23-s + (0.594 − 0.804i)25-s + (−0.574 − 0.818i)27-s + (0.799 + 0.580i)29-s + (−0.321 + 0.233i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61040 - 0.791799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61040 - 0.791799i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.29 + 1.15i)T \) |
| 5 | \( 1 + (-1.99 + 1.00i)T \) |
good | 7 | \( 1 + (-0.814 + 0.814i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.46 + 1.12i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (2.63 - 5.16i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.902 - 0.142i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.29 - 5.91i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.05 + 6.00i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-4.30 - 3.12i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.78 - 1.29i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 1.39i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (2.66 - 0.866i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.01 + 3.01i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.998 - 6.30i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-8.04 + 1.27i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (1.98 + 6.10i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.21 - 12.9i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.976 + 6.16i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (5.88 - 8.09i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (8.26 - 4.21i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-4.29 + 5.90i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 1.20i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (3.41 - 10.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.84 + 0.609i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.94880590084964998528066811117, −10.42683455373756335453561655475, −9.674310201649060481568031054015, −8.681803176430287014223095680449, −7.85594764835072732732184464056, −6.81358693921335150635676205150, −5.69993945477484310402609003345, −4.41095990868933869165892410779, −2.74278767862221416861319339852, −1.53890194684106736944667926933,
2.33381755058840883347216236826, 3.15620262136758452081062859435, 4.99337608660302912415467850116, 5.55151416426510452640395397785, 7.31139741728705109419256886449, 8.026884190770836470668752259068, 9.273304139974456210098885016488, 10.01355291581031987485766608930, 10.57613039972601654545209842508, 11.77095907639718033334410680701