| L(s) = 1 | + (−0.0465 + 0.532i)2-s + (3.65 + 0.645i)4-s + (−4.31 − 2.52i)5-s + (−2.70 − 1.89i)7-s + (−1.06 + 3.97i)8-s + (1.54 − 2.17i)10-s + (6.57 + 2.39i)11-s + (1.00 + 11.5i)13-s + (1.13 − 1.35i)14-s + (11.8 + 4.32i)16-s + (4.62 + 17.2i)17-s + (32.2 − 18.6i)19-s + (−14.1 − 12.0i)20-s + (−1.57 + 3.38i)22-s + (26.7 − 18.7i)23-s + ⋯ |
| L(s) = 1 | + (−0.0232 + 0.266i)2-s + (0.914 + 0.161i)4-s + (−0.862 − 0.505i)5-s + (−0.386 − 0.270i)7-s + (−0.133 + 0.497i)8-s + (0.154 − 0.217i)10-s + (0.597 + 0.217i)11-s + (0.0776 + 0.887i)13-s + (0.0809 − 0.0965i)14-s + (0.743 + 0.270i)16-s + (0.272 + 1.01i)17-s + (1.69 − 0.980i)19-s + (−0.707 − 0.601i)20-s + (−0.0717 + 0.153i)22-s + (1.16 − 0.813i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80511 + 0.489013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80511 + 0.489013i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (4.31 + 2.52i)T \) |
| good | 2 | \( 1 + (0.0465 - 0.532i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (2.70 + 1.89i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-6.57 - 2.39i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 11.5i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-4.62 - 17.2i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-32.2 + 18.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-26.7 + 18.7i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-14.8 - 17.6i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-3.35 + 19.0i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-8.32 - 31.0i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (25.0 + 21.0i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (18.4 + 39.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 3.72i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-40.6 - 40.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-14.6 - 40.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (3.02 + 17.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (1.35 - 0.118i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (49.0 - 84.9i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (19.6 - 73.4i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-39.4 - 47.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (55.9 + 4.89i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-92.2 + 53.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (111. - 51.9i)T + (6.04e3 - 7.20e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.37128603944679574560879476620, −10.30842057493950386860761971421, −9.106560188959357174713643142536, −8.312325393762592012459068338679, −7.12775044582645879199873559261, −6.78235261049705544415212877415, −5.35036863429928792425040215109, −4.10692163799864007210767215141, −3.00399058753608371395369931616, −1.22964411646888844199962795756,
1.02547013741980401686610004198, 2.96184638274349645725688908653, 3.44174939225378679650860487807, 5.24468488371525440286435714164, 6.32905033213784102476271557358, 7.28786766128673916506260184228, 7.921092943401416717500748960292, 9.362782188593326799361335089281, 10.16529361928775812718717357680, 11.13913522703781270239821395950
