| L(s) = 1 | − 3.60·2-s + 15.5i·3-s − 50.9·4-s − 83.0i·5-s − 56.2i·6-s + 414.·8-s − 243·9-s + 299. i·10-s − 442.·11-s − 794. i·12-s − 696. i·13-s + 1.29e3·15-s + 1.76e3·16-s − 6.28e3i·17-s + 876.·18-s + 2.68e3i·19-s + ⋯ |
| L(s) = 1 | − 0.450·2-s + 0.577i·3-s − 0.796·4-s − 0.664i·5-s − 0.260i·6-s + 0.810·8-s − 0.333·9-s + 0.299i·10-s − 0.332·11-s − 0.460i·12-s − 0.317i·13-s + 0.383·15-s + 0.431·16-s − 1.28i·17-s + 0.150·18-s + 0.391i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4459607427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4459607427\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.60T + 64T^{2} \) |
| 5 | \( 1 + 83.0iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 442.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 696. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.28e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.68e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.57e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.32e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.75e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.01e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.51e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 5.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.99e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.87e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.17e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.84e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.75e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.31e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.62e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.75e5iT - 8.32e11T^{2} \) |
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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
−12.38857120407956017820039187357, −11.04025881814645786352424665121, −10.06305210326018076326803115186, −9.145055741742034003582884815676, −8.495693324385542288420924237533, −7.24259923017343720413266210541, −5.31361914642817290317557401158, −4.72400577640711156408702265740, −3.22918040996731432977171231619, −1.11258457351436258812163674787,
0.18985786895099191975885683823, 1.72606567080294662441044475509, 3.37976624399437435148800980670, 4.88074926628060768710945631924, 6.30425461213023573451458089140, 7.44738653455855848515712778237, 8.393783246428457376958175255628, 9.413717494861307922152528381916, 10.52493525293876843016782402312, 11.38485303991059999909803577435
