L(s) = 1 | + (0.0874 − 0.151i)2-s + (5.19 + 0.151i)3-s + (3.98 + 6.90i)4-s + (−2.5 − 4.33i)5-s + (0.477 − 0.773i)6-s + (−4.23 + 7.32i)7-s + 2.79·8-s + (26.9 + 1.57i)9-s − 0.874·10-s + (15.7 − 27.2i)11-s + (19.6 + 36.4i)12-s + (−13.4 − 23.2i)13-s + (0.740 + 1.28i)14-s + (−12.3 − 22.8i)15-s + (−31.6 + 54.7i)16-s − 44.3·17-s + ⋯ |
L(s) = 1 | + (0.0309 − 0.0535i)2-s + (0.999 + 0.0291i)3-s + (0.498 + 0.862i)4-s + (−0.223 − 0.387i)5-s + (0.0324 − 0.0526i)6-s + (−0.228 + 0.395i)7-s + 0.123·8-s + (0.998 + 0.0583i)9-s − 0.0276·10-s + (0.431 − 0.747i)11-s + (0.472 + 0.876i)12-s + (−0.286 − 0.496i)13-s + (0.0141 + 0.0244i)14-s + (−0.212 − 0.393i)15-s + (−0.494 + 0.856i)16-s − 0.632·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.81721 + 0.266036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81721 + 0.266036i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.19 - 0.151i)T \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 2 | \( 1 + (-0.0874 + 0.151i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (4.23 - 7.32i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-15.7 + 27.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (13.4 + 23.2i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 44.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (97.1 + 168. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (1.87 - 3.24i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-125. - 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 62.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-102. - 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-263. + 456. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (77.8 - 134. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-246. - 427. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-379. + 657. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-271. - 470. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 928.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 608.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (307. - 532. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (537. - 931. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-166. + 288. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.52645127040185896794504904376, −14.23740486360877440491890479140, −12.93546138369724728053823854732, −12.19180970826160065294246237581, −10.60792448522987558285264519238, −8.842235804070830816872315385224, −8.182439607014758553411323289523, −6.63624926200621944600628098967, −4.11553204742542410785112687333, −2.58428612889672244095595164725,
2.09297552573748822880765211191, 4.20018360126835296418777212789, 6.49952833781939380312495125318, 7.56891031611451449660741584176, 9.350612851872587895812426769509, 10.26555658111203054151601414585, 11.66130927265110064576032108598, 13.27142800246579590024308655673, 14.36825821764204557617089207470, 15.11045838575390954301676108615