| L(s) = 1 | + 45.1i·3-s + 45.1i·5-s + (−133 + 316. i)7-s − 1.31e3·9-s − 874·11-s − 2.21e3i·13-s − 2.03e3·15-s + 5.96e3i·17-s + 3.11e3i·19-s + (−1.42e4 − 6.00e3i)21-s − 4.73e3·23-s + 1.35e4·25-s − 2.62e4i·27-s + 1.11e4·29-s − 2.74e4i·31-s + ⋯ |
| L(s) = 1 | + 1.67i·3-s + 0.361i·5-s + (−0.387 + 0.921i)7-s − 1.79·9-s − 0.656·11-s − 1.00i·13-s − 0.604·15-s + 1.21i·17-s + 0.454i·19-s + (−1.54 − 0.648i)21-s − 0.389·23-s + 0.869·25-s − 1.33i·27-s + 0.457·29-s − 0.921i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7185793007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7185793007\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (133 - 316. i)T \) |
| good | 3 | \( 1 - 45.1iT - 729T^{2} \) |
| 5 | \( 1 - 45.1iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 874T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.21e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 5.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.11e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 4.73e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.74e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.00e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 5.75e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.14e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.24e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 7.64e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.13e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.95e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.09e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.34e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 6.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 8.14e5iT - 8.32e11T^{2} \) |
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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
−13.13993357787200253256655506065, −11.99264056979823282148928434181, −10.62226182277916047184674247722, −10.24517907146753832095005386178, −9.072811271749236617184909214262, −8.092747967334230954732713323839, −6.06413230432099244801668572873, −5.18053809759035456296527067338, −3.75856884942224993675502341503, −2.68195708644248051985688528437,
0.24021167838274501703095140520, 1.36880019733931573511789669625, 2.84915580985799963762364128930, 4.82689185657972481967586583733, 6.49377077789719751102827339783, 7.16141105188838870396674138524, 8.162791081241336266117612346859, 9.438799918308330870951007759284, 10.94713503559953690667192637801, 12.00907090358046090597276009343
