| L(s) = 1 | + 170.·5-s + 374. i·7-s − 1.35e3i·11-s − 2.95e3·13-s + 6.65e3·17-s − 9.92e3i·19-s − 1.55e4i·23-s + 1.35e4·25-s + 2.08e3·29-s − 2.84e4i·31-s + 6.39e4i·35-s − 3.33e3·37-s + 6.31e4·41-s + 8.85e4i·43-s + 1.08e4i·47-s + ⋯ |
| L(s) = 1 | + 1.36·5-s + 1.09i·7-s − 1.02i·11-s − 1.34·13-s + 1.35·17-s − 1.44i·19-s − 1.28i·23-s + 0.866·25-s + 0.0854·29-s − 0.954i·31-s + 1.49i·35-s − 0.0657·37-s + 0.915·41-s + 1.11i·43-s + 0.104i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.560741155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.560741155\) |
| \(L(4)\) |
|
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 | \( 1 - 170.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 374. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.35e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.95e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 6.65e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.92e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.55e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.08e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.84e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.33e3T + 2.56e9T^{2} \) |
| 41 | \( 1 - 6.31e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 8.85e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.08e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 4.80e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.47e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.24e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 8.75e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.25e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.76e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 4.85e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.06e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.46e6T + 8.32e11T^{2} \) |
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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
−10.52072143778245641247724478020, −9.569531408778018857534510229889, −9.018335223408516908135698978793, −7.83747867208777113482162377733, −6.44407517176598272511578458447, −5.68611343545722017274268894126, −4.86869843176525637838692875116, −2.88346458994246369232403521064, −2.23743653899532241917643801899, −0.64901700315021064173815137994,
1.18042951426613140887910351419, 2.13036245800400425760598257309, 3.62402435092744844403636077152, 4.97159283526502512917589109230, 5.82539274889589546869652028666, 7.12752641874690702743083772687, 7.72561321347055364527812484167, 9.381620460288814863673975850369, 10.04519817984094906344563164343, 10.40751521540254285748908736353
