| L(s) = 1 | − 108.·5-s + 167.·7-s + 1.01e3·11-s − 2.27e3i·13-s + 865. i·17-s + 8.67e3i·19-s + 7.26e3i·23-s − 3.87e3·25-s + 2.10e4·29-s − 1.29e4·31-s − 1.81e4·35-s − 3.25e3i·37-s − 1.12e5i·41-s + 5.93e4i·43-s − 1.17e5i·47-s + ⋯ |
| L(s) = 1 | − 0.867·5-s + 0.488·7-s + 0.763·11-s − 1.03i·13-s + 0.176i·17-s + 1.26i·19-s + 0.596i·23-s − 0.247·25-s + 0.861·29-s − 0.436·31-s − 0.423·35-s − 0.0642i·37-s − 1.63i·41-s + 0.746i·43-s − 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.801834281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801834281\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 108.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 167.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.01e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.27e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 865. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.67e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 7.26e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.10e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.29e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.25e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.12e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.93e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.17e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.17e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 4.70e4T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.52e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.23e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.84e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.04e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.30e4T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.08e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.21e6T + 8.32e11T^{2} \) |
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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
−9.913693865983732744388072156026, −8.753468851427822089091185249293, −8.010459198688669942226706225501, −7.35795141097735660122471851514, −6.15683734299724045297832205835, −5.20515201258782759290983850465, −4.03644322551111839654202398778, −3.35740533047340603111658845657, −1.83790109681372042907887412544, −0.70178108980415732381576943596,
0.54888298404768999290385266200, 1.74813475941834480004270118504, 3.05555904013141675131652452004, 4.25887349059622319683334719934, 4.77389820107009108755513332214, 6.28335169054861555935731038938, 7.04020134169021008726929922174, 7.961566080442308605028771270850, 8.832298634233921864974014674924, 9.550659706756110653090807190452
