| L(s) = 1 | + (−0.342 + 1.97i)2-s + (−3.76 − 1.34i)4-s + 11.5i·7-s + (3.94 − 6.95i)8-s + 9.97i·11-s + 14.1·13-s + (−22.6 − 3.94i)14-s + (12.3 + 10.1i)16-s + 30.5·17-s − 12.2i·19-s + (−19.6 − 3.41i)22-s + 15.7i·23-s + (−4.83 + 27.8i)26-s + (15.5 − 43.3i)28-s + 18.8·29-s + ⋯ |
| L(s) = 1 | + (−0.171 + 0.985i)2-s + (−0.941 − 0.337i)4-s + 1.64i·7-s + (0.493 − 0.869i)8-s + 0.906i·11-s + 1.08·13-s + (−1.62 − 0.281i)14-s + (0.772 + 0.635i)16-s + 1.79·17-s − 0.643i·19-s + (−0.893 − 0.155i)22-s + 0.685i·23-s + (−0.185 + 1.07i)26-s + (0.554 − 1.54i)28-s + 0.651·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.549218869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549218869\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 1.97i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 11.5iT - 49T^{2} \) |
| 11 | \( 1 - 9.97iT - 121T^{2} \) |
| 13 | \( 1 - 14.1T + 169T^{2} \) |
| 17 | \( 1 - 30.5T + 289T^{2} \) |
| 19 | \( 1 + 12.2iT - 361T^{2} \) |
| 23 | \( 1 - 15.7iT - 529T^{2} \) |
| 29 | \( 1 - 18.8T + 841T^{2} \) |
| 31 | \( 1 + 35.2iT - 961T^{2} \) |
| 37 | \( 1 + 50.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 55.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.46T + 2.80e3T^{2} \) |
| 59 | \( 1 - 64.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 30.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 2.24T + 5.32e3T^{2} \) |
| 79 | \( 1 - 78.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 146. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 87.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 126.T + 9.40e3T^{2} \) |
| 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
−9.842687602170467614221672427728, −9.370515729630108941895813042885, −8.504659973819015975610692996234, −7.84533114845665656943095484321, −6.85352591281773344118502864184, −5.79494857702572218402012673499, −5.45513576406465477761701318074, −4.25177199998870445865741685785, −2.94075335660300117486023935485, −1.37997079264051228435563567093,
0.63803368204607031436974657715, 1.45045907740180996795467612935, 3.37191898943700935933245370934, 3.63175253726230393222114365661, 4.84999272062362422926296044927, 5.95940256566620490901126684055, 7.17282609395544979654058133545, 8.101189290700376638617980243785, 8.673328817184199583856695109450, 9.896595759160484786939562930831
