L(s) = 1 | + (0.0847 − 0.0489i)2-s + (−3.99 + 6.91i)4-s + (−9.06 + 15.6i)5-s + (12.7 − 13.4i)7-s + 1.56i·8-s + 1.77i·10-s + (−32.0 + 18.5i)11-s + (−16.3 − 9.44i)13-s + (0.423 − 1.76i)14-s + (−31.8 − 55.2i)16-s + 62.5·17-s − 70.2i·19-s + (−72.4 − 125. i)20-s + (−1.81 + 3.13i)22-s + (−140. − 81.2i)23-s + ⋯ |
L(s) = 1 | + (0.0299 − 0.0173i)2-s + (−0.499 + 0.864i)4-s + (−0.810 + 1.40i)5-s + (0.688 − 0.725i)7-s + 0.0691i·8-s + 0.0561i·10-s + (−0.878 + 0.507i)11-s + (−0.349 − 0.201i)13-s + (0.00808 − 0.0336i)14-s + (−0.498 − 0.862i)16-s + 0.891·17-s − 0.848i·19-s + (−0.809 − 1.40i)20-s + (−0.0175 + 0.0304i)22-s + (−1.27 − 0.736i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0774906 - 0.271083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0774906 - 0.271083i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-12.7 + 13.4i)T \) |
good | 2 | \( 1 + (-0.0847 + 0.0489i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (9.06 - 15.6i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (32.0 - 18.5i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (16.3 + 9.44i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 62.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (140. + 81.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (82.5 - 47.6i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-110. - 63.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (99.1 - 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-160. - 278. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (79.3 + 137. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 191. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-106. + 185. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (190. - 110. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-68.2 + 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 458. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 967. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-298. - 516. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (180. + 313. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 35.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.15e3 - 664. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.55620867544473611538751224081, −11.71083053544357790127365634530, −10.75082409613450736913782859798, −9.973848682018302525208606200515, −8.246864305504514674140130341131, −7.63683335026882960892367504870, −6.89695272034969419883494382293, −4.91875134882105967444264652458, −3.80258043525853445445955385545, −2.70532339301319941478123919327,
0.12463289292330774072249439260, 1.64743563513877180528072955557, 3.99508966392389184354531847551, 5.19698477428264103380093404855, 5.66796541947869906134596819101, 7.82945767171211239389004573620, 8.438301721357631277089540919487, 9.410483664572936524661454163392, 10.47774529506290044784177843381, 11.79819576755867518575176076347