L(s) = 1 | − 27i·3-s + 344. i·5-s + 1.66e3·7-s − 729·9-s − 2.28e3i·11-s − 1.15e4i·13-s + 9.29e3·15-s − 1.72e4·17-s + 1.45e4i·19-s − 4.50e4i·21-s − 4.08e4·23-s − 4.04e4·25-s + 1.96e4i·27-s − 1.59e5i·29-s − 1.71e5·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.23i·5-s + 1.83·7-s − 0.333·9-s − 0.516i·11-s − 1.45i·13-s + 0.711·15-s − 0.851·17-s + 0.486i·19-s − 1.06i·21-s − 0.700·23-s − 0.517·25-s + 0.192i·27-s − 1.21i·29-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.270000448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270000448\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
good | 5 | \( 1 - 344. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 1.66e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.28e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 1.15e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.72e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.45e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 4.08e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.59e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.71e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.02e3iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 7.35e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.80e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.40e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 7.27e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.21e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.59e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.24e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.38e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 9.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.87e6T + 8.07e13T^{2} \) |
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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
−10.09627784869820821844028364721, −8.455666369461534378065982069564, −7.980972505984529575266834433039, −7.12519426516757439826858499846, −6.01974553457459229284392152303, −5.12387181280276676567973909334, −3.71167811640470043758541590768, −2.51664713737584135192155192956, −1.62655948264052987946366071035, −0.23967245070304361214897837582,
1.37777400558786488045057717935, 2.02446455628715034609537460342, 4.04201053557396057561832850917, 4.76539509637654389676692936519, 5.19631202689642882923646632471, 6.79785150636743511945135127504, 7.984377627800964289495468125576, 8.805687759330101094139129291724, 9.275137284103501731356583097699, 10.60600220562858540037652576811