| L(s) = 1 | + 3.85i·2-s + 13.0·3-s + 49.1·4-s − 100.·5-s + 50.4i·6-s − 365.·7-s + 435. i·8-s − 557.·9-s − 386. i·10-s − 61.8i·11-s + 643.·12-s + 4.00e3i·13-s − 1.40e3i·14-s − 1.31e3·15-s + 1.46e3·16-s − 670.·17-s + ⋯ |
| L(s) = 1 | + 0.481i·2-s + 0.485·3-s + 0.768·4-s − 0.803·5-s + 0.233i·6-s − 1.06·7-s + 0.851i·8-s − 0.764·9-s − 0.386i·10-s − 0.0464i·11-s + 0.372·12-s + 1.82i·13-s − 0.512i·14-s − 0.389·15-s + 0.358·16-s − 0.136·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.132780 + 0.967121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132780 + 0.967121i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-1.97e5 - 5.53e4i)T \) |
| good | 2 | \( 1 - 3.85iT - 64T^{2} \) |
| 3 | \( 1 - 13.0T + 729T^{2} \) |
| 5 | \( 1 + 100.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 365.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 61.8iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 4.00e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 670.T + 2.41e7T^{2} \) |
| 19 | \( 1 + 4.25e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 6.23e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 741.T + 5.94e8T^{2} \) |
| 31 | \( 1 + 4.93e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 5.81e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.74e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.06e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.99e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.64e5T + 2.21e10T^{2} \) |
| 61 | \( 1 + 1.01e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.20e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.38e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + 4.94e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.58e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.30e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.83e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.79e5iT - 8.32e11T^{2} \) |
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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
−14.61815418270711665977490740149, −13.48524597445712745645536683138, −11.92421758326211646021255157814, −11.25583198196343858395581654613, −9.513694460489204898858328196774, −8.311665609679366652412552271739, −7.09207423306693249364856032490, −6.05766563909247107582226736146, −3.88377926318059913488992404368, −2.38613479711768621972856128922,
0.34625703794940649380845418517, 2.69564030838615853252319213102, 3.56644923458120810266503398676, 5.93577722657179614750357149510, 7.36099560662932445700476211223, 8.547105027619275888114772629513, 10.10038869451754200681223178827, 11.04950775133663448587466865191, 12.30737898534643674901272160082, 13.02618729972761385941363995547
