| L(s) = 1 | + 33.1i·3-s + 225.·5-s + 256.·7-s − 370.·9-s + 1.58e3·11-s + 516. i·13-s + 7.49e3i·15-s − 8.57e3·17-s + (34.9 + 6.85e3i)19-s + 8.49e3i·21-s + 4.35e3·23-s + 3.54e4·25-s + 1.18e4i·27-s − 4.12e4i·29-s + 1.03e4i·31-s + ⋯ |
| L(s) = 1 | + 1.22i·3-s + 1.80·5-s + 0.747·7-s − 0.508·9-s + 1.19·11-s + 0.235i·13-s + 2.22i·15-s − 1.74·17-s + (0.00509 + 0.999i)19-s + 0.917i·21-s + 0.358·23-s + 2.26·25-s + 0.603i·27-s − 1.69i·29-s + 0.347i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00509 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.00509 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.743633535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743633535\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-34.9 - 6.85e3i)T \) |
| good | 3 | \( 1 - 33.1iT - 729T^{2} \) |
| 5 | \( 1 - 225.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 256.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.58e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 516. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 8.57e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 4.35e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.12e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.03e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.35e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 7.85e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.39e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 4.15e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.20e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.04e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.24e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.77e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.49e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.47e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.98e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 9.64e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 6.88e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.19e6iT - 8.32e11T^{2} \) |
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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
−10.76714455799656236891630638072, −9.792624458114684796253754309576, −9.369873550377869878417254052783, −8.492206026332951710954082616628, −6.72336668012601988579075826155, −5.89971792587156346689561958572, −4.82002311969866112549142688188, −4.03039322049780636716548281643, −2.35184432248399716489808799591, −1.38368433166663266597336811839,
0.941657029972780002235119765162, 1.76171746852597521975605569129, 2.50864817805972123172793283667, 4.56306076128193853750692801359, 5.74445596901609933104054266108, 6.63269845183073351595115035762, 7.20883962132413012705396932283, 8.827970575316696855647204871933, 9.172646150558765400475460244485, 10.58384436237778226193558414790
