| L(s) = 1 | + (0.183 − 1.15i)2-s + (−1.68 + 0.858i)3-s + (13.9 + 4.51i)4-s + (19.3 − 15.8i)5-s + (0.685 + 2.10i)6-s + (28.2 − 28.2i)7-s + (16.3 − 32.0i)8-s + (−45.5 + 62.6i)9-s + (−14.7 − 25.3i)10-s + (−46.0 + 33.4i)11-s + (−27.3 + 4.32i)12-s + (15.2 + 96.1i)13-s + (−27.5 − 37.9i)14-s + (−19.0 + 43.2i)15-s + (155. + 112. i)16-s + (−467. − 237. i)17-s + ⋯ |
| L(s) = 1 | + (0.0458 − 0.289i)2-s + (−0.187 + 0.0953i)3-s + (0.869 + 0.282i)4-s + (0.774 − 0.632i)5-s + (0.0190 + 0.0585i)6-s + (0.576 − 0.576i)7-s + (0.254 − 0.500i)8-s + (−0.561 + 0.773i)9-s + (−0.147 − 0.253i)10-s + (−0.380 + 0.276i)11-s + (−0.189 + 0.0300i)12-s + (0.0901 + 0.569i)13-s + (−0.140 − 0.193i)14-s + (−0.0847 + 0.192i)15-s + (0.606 + 0.440i)16-s + (−1.61 − 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.58772 - 0.314140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58772 - 0.314140i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-19.3 + 15.8i)T \) |
| good | 2 | \( 1 + (-0.183 + 1.15i)T + (-15.2 - 4.94i)T^{2} \) |
| 3 | \( 1 + (1.68 - 0.858i)T + (47.6 - 65.5i)T^{2} \) |
| 7 | \( 1 + (-28.2 + 28.2i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + (46.0 - 33.4i)T + (4.52e3 - 1.39e4i)T^{2} \) |
| 13 | \( 1 + (-15.2 - 96.1i)T + (-2.71e4 + 8.82e3i)T^{2} \) |
| 17 | \( 1 + (467. + 237. i)T + (4.90e4 + 6.75e4i)T^{2} \) |
| 19 | \( 1 + (382. - 124. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + (-493. - 78.1i)T + (2.66e5 + 8.64e4i)T^{2} \) |
| 29 | \( 1 + (209. + 68.1i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-470. - 1.44e3i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (1.34e3 - 212. i)T + (1.78e6 - 5.79e5i)T^{2} \) |
| 41 | \( 1 + (688. + 499. i)T + (8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + (-143. - 143. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (223. + 437. i)T + (-2.86e6 + 3.94e6i)T^{2} \) |
| 53 | \( 1 + (-1.14e3 + 585. i)T + (4.63e6 - 6.38e6i)T^{2} \) |
| 59 | \( 1 + (-3.95e3 + 5.43e3i)T + (-3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-2.49e3 + 1.81e3i)T + (4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 + (-2.04e3 - 1.04e3i)T + (1.18e7 + 1.63e7i)T^{2} \) |
| 71 | \( 1 + (-670. + 2.06e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (6.87e3 + 1.08e3i)T + (2.70e7 + 8.77e6i)T^{2} \) |
| 79 | \( 1 + (-7.99e3 - 2.59e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (1.92e3 - 3.78e3i)T + (-2.78e7 - 3.83e7i)T^{2} \) |
| 89 | \( 1 + (4.49e3 + 6.18e3i)T + (-1.93e7 + 5.96e7i)T^{2} \) |
| 97 | \( 1 + (3.15e3 + 6.19e3i)T + (-5.20e7 + 7.16e7i)T^{2} \) |
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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
−16.84364685481917119026347385019, −15.75756891255443495941117845196, −14.02123384115343898614198914391, −12.90912182855274460663863206301, −11.38698845253131375578570308186, −10.45488847179416458556636079755, −8.565904504076937037349392894264, −6.81285866591491934283932949608, −4.85712132796698637200257571752, −2.07536881238256304250543520642,
2.39358125832714963162679926674, 5.68383371835788159555438211622, 6.69924250402433229354087478668, 8.658612993986354438065111717047, 10.59361183049225637033797076646, 11.45743929273378529082191594588, 13.17976191311878123727016939286, 14.87429833956615868345847022806, 15.23553081407941956281317002358, 17.09807991755522348758167590460
