| L(s) = 1 | + (−1 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (3.43 + 5.95i)5-s − 6·6-s + (0.809 − 18.5i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (6.87 − 11.9i)10-s + (−18.6 + 32.3i)11-s + (6.00 + 10.3i)12-s + 13·13-s + (−32.8 + 17.1i)14-s + 20.6·15-s + (−8 − 13.8i)16-s + (−47.4 + 82.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.307 + 0.532i)5-s − 0.408·6-s + (0.0437 − 0.999i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.217 − 0.376i)10-s + (−0.512 + 0.886i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.627 + 0.326i)14-s + 0.354·15-s + (−0.125 − 0.216i)16-s + (−0.676 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08381610005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08381610005\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 + (-0.809 + 18.5i)T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + (-3.43 - 5.95i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (18.6 - 32.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (47.4 - 82.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (52.7 + 91.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (18.5 + 32.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 57.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + (111. - 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-13.7 - 23.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 495.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (232. + 402. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (136. - 236. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (144. - 249. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (99.3 + 172. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (103. - 179. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (162. - 281. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-529. - 917. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 849.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-692. - 1.19e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 150.T + 9.12e5T^{2} \) |
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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
−10.58463209205563521926785461910, −10.05913700683047253232265793273, −8.886979602594535820839765917545, −8.094172954605367139692162929939, −7.05632132405676577197226228694, −6.50876875996685753366836484407, −4.81297090560770657818078459884, −3.76443068629937468415470493447, −2.52777409527052387920842458033, −1.55151364233386196288286391606,
0.02584923422611186985743498030, 1.86181347333188988082796800062, 3.23387360586726140856530069563, 4.69332465179365634390097575260, 5.54400052877536735166770770749, 6.23752279434939814634398025677, 7.68861633853041971601697976280, 8.473292587077232752978640136936, 9.135796894926460834127920286850, 9.757433917905493803371085946403