| L(s) = 1 | − 6.89i·3-s + (10.7 − 2.89i)5-s + 12.6i·7-s − 20.5·9-s − 59.1·11-s − 42.2i·13-s + (−20 − 74.4i)15-s − 126. i·17-s − 19.1·19-s + 87.5·21-s + 78.3i·23-s + (108. − 62.6i)25-s − 44.1i·27-s − 148.·29-s − 139.·31-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s + (0.965 − 0.259i)5-s + 0.685i·7-s − 0.762·9-s − 1.62·11-s − 0.900i·13-s + (−0.344 − 1.28i)15-s − 1.80i·17-s − 0.231·19-s + 0.910·21-s + 0.709i·23-s + (0.865 − 0.500i)25-s − 0.314i·27-s − 0.950·29-s − 0.806·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.387594122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387594122\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-10.7 + 2.89i)T \) |
| good | 3 | \( 1 + 6.89iT - 27T^{2} \) |
| 7 | \( 1 - 12.6iT - 343T^{2} \) |
| 11 | \( 1 + 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 126. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 66.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 288. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 360. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 686. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 83.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 208.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 122. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 573. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 643. iT - 9.12e5T^{2} \) |
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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
−10.80632782452705642907278579056, −9.789824299673142417024347609020, −8.789792759336213004315583879361, −7.73694661668826087774957016489, −7.03527839650686385237104183183, −5.63160974561454941839569233864, −5.29172547399707346140844087125, −2.81209772901212244713339272471, −2.02489550531231270614042593216, −0.46469174309031608193399381287,
1.99446929662126804942197754958, 3.49562807113466535314530068425, 4.56494430609073835159457240524, 5.52386615201605147612056556416, 6.62406624286703610929079267306, 7.974389790177209400240482610961, 9.089610143459325957297189923613, 10.00331660309603203122933819677, 10.55393140032480418634423908995, 11.05252885641099209065578839370
