L(s) = 1 | + 2.23i·3-s + 2.82·7-s − 2.00·9-s − 2.23i·11-s − 6.32i·13-s + 5·17-s + 2.23i·19-s + 6.32i·21-s − 5.65·23-s + 2.23i·27-s − 6.32i·29-s + 5.00·33-s − 6.32i·37-s + 14.1·39-s + 3·41-s + ⋯ |
L(s) = 1 | + 1.29i·3-s + 1.06·7-s − 0.666·9-s − 0.674i·11-s − 1.75i·13-s + 1.21·17-s + 0.512i·19-s + 1.38i·21-s − 1.17·23-s + 0.430i·27-s − 1.17i·29-s + 0.870·33-s − 1.03i·37-s + 2.26·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.067602172\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.067602172\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 6.32iT - 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 6.32iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.32iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 6.32iT - 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.528235648593454946915794748364, −8.062857124344856400936099975180, −7.51065414602691577555572046872, −5.99950637814402003534263259890, −5.49282252838308795576768562759, −4.90696250564629571546894774338, −3.83133781389755788404528064438, −3.43070761547423316832095447571, −2.15258750165526804654800071894, −0.68537427364798018976735953965,
1.29827131767529395022721206120, 1.73106325099570893526123908111, 2.72694223950730905013283491281, 4.15618811157245201920897066324, 4.75523619421423959895978080407, 5.79677986038035335978775471115, 6.60387982229154163232832969316, 7.23788856285094856755055337985, 7.78036994404254264475300009536, 8.439690379984450725001387532723