L(s) = 1 | + (0.923 − 0.382i)2-s + (0.253 + 1.27i)3-s + (0.707 − 0.707i)4-s + (0.580 − 2.15i)5-s + (0.722 + 1.08i)6-s + (0.192 − 0.128i)7-s + (0.382 − 0.923i)8-s + (1.21 − 0.501i)9-s + (−0.289 − 2.21i)10-s + (−3.12 + 2.08i)11-s + (1.08 + 0.722i)12-s + 6.31i·13-s + (0.128 − 0.192i)14-s + (2.89 + 0.192i)15-s − i·16-s + (3.51 − 2.15i)17-s + ⋯ |
L(s) = 1 | + (0.653 − 0.270i)2-s + (0.146 + 0.735i)3-s + (0.353 − 0.353i)4-s + (0.259 − 0.965i)5-s + (0.294 + 0.441i)6-s + (0.0728 − 0.0486i)7-s + (0.135 − 0.326i)8-s + (0.403 − 0.167i)9-s + (−0.0916 − 0.701i)10-s + (−0.941 + 0.629i)11-s + (0.311 + 0.208i)12-s + 1.75i·13-s + (0.0344 − 0.0514i)14-s + (0.748 + 0.0497i)15-s − 0.250i·16-s + (0.852 − 0.522i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71065 - 0.148526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71065 - 0.148526i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 + 0.382i)T \) |
| 5 | \( 1 + (-0.580 + 2.15i)T \) |
| 17 | \( 1 + (-3.51 + 2.15i)T \) |
good | 3 | \( 1 + (-0.253 - 1.27i)T + (-2.77 + 1.14i)T^{2} \) |
| 7 | \( 1 + (-0.192 + 0.128i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (3.12 - 2.08i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 - 6.31iT - 13T^{2} \) |
| 19 | \( 1 + (5.28 + 2.18i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (8.43 + 1.67i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (0.221 + 1.11i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (1.28 + 0.856i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (0.247 - 0.0492i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.06 + 5.36i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-8.89 - 3.68i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 + (-2.39 - 5.77i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.547 - 1.32i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (10.1 + 2.02i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.973 + 0.973i)T - 67iT^{2} \) |
| 71 | \( 1 + (1.06 - 1.59i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-1.60 - 1.07i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.466 + 0.698i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-13.3 + 5.53i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-9.96 + 9.96i)T - 89iT^{2} \) |
| 97 | \( 1 + (-4.18 - 2.79i)T + (37.1 + 89.6i)T^{2} \) |
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
−12.62764510511899939165543171425, −12.04273872499916010790756671147, −10.68853999343415845009980895732, −9.768083261462818911983217396228, −9.009134223084019916388870111007, −7.51521254161019686196283601760, −6.04904405664183390491695842673, −4.66059222779125805278227774585, −4.18002954264777240145140164556, −2.08396029103170665911535333307,
2.31008053851880225259882056984, 3.61033427811202508899576615377, 5.54145533666023185838440123694, 6.29881528454507597046184350376, 7.73185893347043367412004720953, 8.028488197970287879209516508806, 10.25525982618641936150601265393, 10.65932857606861697339661781438, 12.18191204840209327415268711432, 12.94824733565806980536761123258