| L(s) = 1 | + 15.5·3-s − 103. i·5-s − 108i·7-s + 243·9-s + 394.·11-s + 2.20e3i·13-s − 1.62e3i·15-s + 974·17-s + 976.·19-s − 1.68e3i·21-s + 2.09e4i·23-s + 4.82e3·25-s + 3.78e3·27-s − 9.49e3i·29-s + 1.56e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.831i·5-s − 0.314i·7-s + 0.333·9-s + 0.296·11-s + 1.00i·13-s − 0.479i·15-s + 0.198·17-s + 0.142·19-s − 0.181i·21-s + 1.72i·23-s + 0.308·25-s + 0.192·27-s − 0.389i·29-s + 0.525i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.866045888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.866045888\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5T \) |
| good | 5 | \( 1 + 103. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 108iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 394.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.20e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 974T + 2.41e7T^{2} \) |
| 19 | \( 1 - 976.T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.09e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 9.49e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.56e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 5.14e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 3.32e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.63e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.32e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.64e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 7.50e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 4.82e3iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.61e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.65e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.13e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.58e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.76e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.64e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 5.16e4T + 8.32e11T^{2} \) |
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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
−10.09314162721794652677688372121, −9.305118482781837309387556785524, −8.621505282772745765427750927726, −7.61519836892159543668483578916, −6.69566023616122601877477426754, −5.36610243705979988651535816864, −4.35424358274137547603134643390, −3.40097561072295266849914413708, −1.89545371936674483608457847456, −0.936839107106656775970695514251,
0.74780355906084811322795620247, 2.34977284611046894645613867236, 3.08380121660886981772636107745, 4.25853908931475667710260078176, 5.61667677396050749782403784789, 6.64214597052214642252139018581, 7.56671815129007911805069136495, 8.476232436964147984905703753001, 9.384851358812852770956648138355, 10.43002872875736855073694877719
