| L(s) = 1 | + (1.59 − 2.76i)2-s + (0.340 − 8.99i)3-s + (2.90 + 5.03i)4-s + (6.24 − 24.2i)5-s + (−24.3 − 15.2i)6-s + (−21.0 − 12.1i)7-s + 69.6·8-s + (−80.7 − 6.11i)9-s + (−56.9 − 55.8i)10-s + (−35.8 − 20.6i)11-s + (46.2 − 24.4i)12-s + (−78.1 + 45.1i)13-s + (−67.3 + 38.8i)14-s + (−215. − 64.3i)15-s + (64.6 − 111. i)16-s + 318.·17-s + ⋯ |
| L(s) = 1 | + (0.399 − 0.691i)2-s + (0.0377 − 0.999i)3-s + (0.181 + 0.314i)4-s + (0.249 − 0.968i)5-s + (−0.675 − 0.424i)6-s + (−0.430 − 0.248i)7-s + 1.08·8-s + (−0.997 − 0.0755i)9-s + (−0.569 − 0.558i)10-s + (−0.295 − 0.170i)11-s + (0.321 − 0.169i)12-s + (−0.462 + 0.267i)13-s + (−0.343 + 0.198i)14-s + (−0.958 − 0.286i)15-s + (0.252 − 0.437i)16-s + 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.991032 - 1.67545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991032 - 1.67545i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.340 + 8.99i)T \) |
| 5 | \( 1 + (-6.24 + 24.2i)T \) |
| good | 2 | \( 1 + (-1.59 + 2.76i)T + (-8 - 13.8i)T^{2} \) |
| 7 | \( 1 + (21.0 + 12.1i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (35.8 + 20.6i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (78.1 - 45.1i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 318.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 572.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-93.6 - 162. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-1.20e3 - 694. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-91.5 - 158. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 348. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.88e3 - 1.09e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.19e3 + 689. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-342. + 592. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.09e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.76e3 + 2.75e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.22e3 - 5.58e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (4.14e3 - 2.39e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 6.78e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.04e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.10e3 - 3.64e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (3.10e3 - 5.38e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 1.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.91e3 + 1.10e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−14.05696417571944701748698348451, −13.30220264997998414170652122113, −12.34192623984645893380307446530, −11.70822202214320697719480195447, −9.955534704914306938860329356240, −8.294812227822167940914762644454, −7.11496818350644476038608675196, −5.23442076622545951279622158122, −3.11969393848671551659703958273, −1.27552587183774761481507248510,
3.04483935199105213382458768917, 5.07089353947738195467708128247, 6.20472400944704906021785481574, 7.67503826402669440707485146659, 9.778066375019022959539555363968, 10.36324394178061823106593908818, 11.75846153549530335949115256408, 13.75273071800790301404039260758, 14.53139506304560363562120130053, 15.42639911049211692652783572066
