| L(s) = 1 | + (−22.7 − 39.4i)3-s + (−1.50e3 − 866. i)5-s + (6.34e3 − 185. i)7-s + (8.80e3 − 1.52e4i)9-s + (7.18e4 − 4.14e4i)11-s + 1.51e5i·13-s + 7.89e4i·15-s + (−3.25e5 + 1.87e5i)17-s + (−1.67e5 + 2.89e5i)19-s + (−1.51e5 − 2.46e5i)21-s + (1.23e6 + 7.11e5i)23-s + (5.24e5 + 9.08e5i)25-s − 1.69e6·27-s + 5.11e6·29-s + (4.43e5 + 7.67e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.162 − 0.281i)3-s + (−1.07 − 0.619i)5-s + (0.999 − 0.0291i)7-s + (0.447 − 0.774i)9-s + (1.47 − 0.854i)11-s + 1.47i·13-s + 0.402i·15-s + (−0.944 + 0.545i)17-s + (−0.294 + 0.510i)19-s + (−0.170 − 0.276i)21-s + (0.917 + 0.529i)23-s + (0.268 + 0.465i)25-s − 0.615·27-s + 1.34·29-s + (0.0861 + 0.149i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.023517672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023517672\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.34e3 + 185. i)T \) |
| good | 3 | \( 1 + (22.7 + 39.4i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.50e3 + 866. i)T + (9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-7.18e4 + 4.14e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.51e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (3.25e5 - 1.87e5i)T + (5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.67e5 - 2.89e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.23e6 - 7.11e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 5.11e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-4.43e5 - 7.67e5i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-7.46e6 + 1.29e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.60e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 2.69e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.88e7 + 5.00e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.55e7 - 7.89e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.65e6 - 2.86e6i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.99e7 + 4.61e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-4.61e7 + 2.66e7i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.85e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.17e8 + 1.25e8i)T + (2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (3.29e8 + 1.90e8i)T + (5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 6.02e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-3.63e8 - 2.09e8i)T + (1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 2.18e8iT - 7.60e17T^{2} \) |
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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
−11.77130091817704914478030515295, −11.01727204427580223295003315947, −9.105712282619263698238522477352, −8.588237953890072531136755187919, −7.23026376152261503609928867974, −6.23604136981789749494178526672, −4.41904657589343293238511315315, −3.90098423680623259506018620973, −1.61621515166412455991917235605, −0.71634848897631918030816988723,
0.984504635588945826783722408684, 2.61286087625931878865910145580, 4.18751447111175402237115231728, 4.89445999672860280064004161576, 6.76012404201219569284280066086, 7.63139000676005036806149375705, 8.664951482610168678873950629364, 10.17165933337837000842845209120, 11.10630442515314200321734976190, 11.73542166395241867439891691585
