L(s) = 1 | + 527.·5-s + 1.31e3i·7-s + 961. i·11-s − 1.07e4i·13-s + 1.15e4i·17-s + 4.69e4·19-s − 8.87e3·23-s + 2.00e5·25-s + 1.09e5·29-s − 2.39e5i·31-s + 6.92e5i·35-s + 3.66e5i·37-s − 4.07e4i·41-s − 2.25e5·43-s + 7.13e5·47-s + ⋯ |
L(s) = 1 | + 1.88·5-s + 1.44i·7-s + 0.217i·11-s − 1.35i·13-s + 0.569i·17-s + 1.56·19-s − 0.152·23-s + 2.56·25-s + 0.833·29-s − 1.44i·31-s + 2.73i·35-s + 1.18i·37-s − 0.0922i·41-s − 0.432·43-s + 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.596243868\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.596243868\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 527.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.31e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 961. iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 1.07e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.15e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 4.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.87e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.39e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.66e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 4.07e4iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 2.25e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.86e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.19e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.03e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.27e4T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.92e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.21e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 1.39e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.64e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.75e5T + 8.07e13T^{2} \) |
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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
−10.36575096441178360930897275427, −9.752308913229782425355864536606, −8.991583688471375116553510631093, −7.952125422415092138684012374313, −6.39925191708936043742155905670, −5.67027661594875455322680430221, −5.12638300475092395459989677477, −3.00503997036661003374184974077, −2.25851832449021024009746678933, −1.12020328301664998411493462701,
0.909849497386132744181249836483, 1.74019033596148943375809183558, 3.07732470469705044714126707924, 4.51602157584001795409872935874, 5.52471066842984309705530038942, 6.65170066095711567916201119240, 7.27366126215788853213119180895, 8.890125973965450931180947548375, 9.662494964146284863272304102739, 10.30315757556548975624287643256