| L(s) = 1 | + 5.65i·2-s − 32.0·4-s + 484·7-s − 181. i·8-s − 1.34e3i·11-s − 3.36e3·13-s + 2.73e3i·14-s + 1.02e3·16-s + 12.7i·17-s + 5.74e3·19-s + 7.58e3·22-s − 3.37e3i·23-s − 1.90e4i·26-s − 1.54e4·28-s − 2.93e4i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.41·7-s − 0.353i·8-s − 1.00i·11-s − 1.53·13-s + 0.997i·14-s + 0.250·16-s + 0.00259i·17-s + 0.837·19-s + 0.712·22-s − 0.277i·23-s − 1.08i·26-s − 0.705·28-s − 1.20i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.01037654529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01037654529\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 484T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.36e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.37e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.93e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.97e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.80e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 7.67e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.32e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.68e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.51e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.09e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.21e5T + 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.570777892166045924374751290871, −8.651751680574406803009365971516, −7.79532032297496981636789602243, −7.21255387840803199088719846017, −5.82255638763566398631356230058, −5.10858422290881220503771753251, −4.19280308969180733019412290320, −2.70822628848661082135491629480, −1.31597678605706912600317125889, −0.00215436408347798118770047268,
1.53332702923560027312757993144, 2.19950392581183444509625105230, 3.60136241162053389919335700500, 4.97246406253046099714689393971, 5.12756626901347607236435244318, 7.10936171125943187962814236950, 7.69198312777834482587600532390, 8.821400692493894137392877716640, 9.700944834260700688310236305897, 10.49648953821918090087594765258
