L(s) = 1 | + (1.5 − 2.59i)3-s + (14.9 − 8.63i)5-s + (15.3 + 10.3i)7-s + (−4.5 − 7.79i)9-s + (−35.5 − 20.5i)11-s + 7.35i·13-s − 51.8i·15-s + (−98.9 − 57.1i)17-s + (−72.7 − 126. i)19-s + (49.9 − 24.2i)21-s + (−181. + 104. i)23-s + (86.5 − 149. i)25-s − 27·27-s + 136.·29-s + (62.3 − 108. i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (1.33 − 0.772i)5-s + (0.827 + 0.561i)7-s + (−0.166 − 0.288i)9-s + (−0.973 − 0.562i)11-s + 0.156i·13-s − 0.891i·15-s + (−1.41 − 0.814i)17-s + (−0.878 − 1.52i)19-s + (0.519 − 0.251i)21-s + (−1.64 + 0.948i)23-s + (0.692 − 1.19i)25-s − 0.192·27-s + 0.873·29-s + (0.361 − 0.625i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.156999526\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.156999526\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 + (-15.3 - 10.3i)T \) |
good | 5 | \( 1 + (-14.9 + 8.63i)T + (62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (35.5 + 20.5i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 7.35iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (98.9 + 57.1i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (72.7 + 126. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (181. - 104. i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-62.3 + 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (141. + 244. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 208. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 232. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-318. - 552. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-221. + 382. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (111. - 192. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-436. + 251. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-413. - 238. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 58.6iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-182. - 105. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (25.7 - 14.8i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 516.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-543. + 313. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 609. iT - 9.12e5T^{2} \) |
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
−9.546373221625589033067001426952, −8.845598413207339173675526862226, −8.318322662921837793053120411644, −7.14503006330486605084077401923, −6.03370664616517620754849802696, −5.34706569011045748967792342567, −4.44415073757347374383591561453, −2.41851029682351621686082657540, −2.08462071605653377314358781132, −0.51325863833596712369510161543,
1.83414782410225819324911478851, 2.45535272421425956006560647491, 3.98869306352075404710566925531, 4.88008923634610482153060932600, 6.01184548179763995076726291181, 6.72694042209102042408131824478, 8.042023383852030357311943024377, 8.555398741752235747905681119073, 10.02045437664698952931582244988, 10.33063457621971528452452024484