L(s) = 1 | − 2-s + (0.482 + 1.66i)3-s + 4-s + 0.885i·5-s + (−0.482 − 1.66i)6-s − i·7-s − 8-s + (−2.53 + 1.60i)9-s − 0.885i·10-s + (1.14 + 3.11i)11-s + (0.482 + 1.66i)12-s + 1.32i·13-s + i·14-s + (−1.47 + 0.426i)15-s + 16-s − 1.35·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.278 + 0.960i)3-s + 0.5·4-s + 0.395i·5-s + (−0.196 − 0.679i)6-s − 0.377i·7-s − 0.353·8-s + (−0.845 + 0.534i)9-s − 0.279i·10-s + (0.345 + 0.938i)11-s + (0.139 + 0.480i)12-s + 0.368i·13-s + 0.267i·14-s + (−0.380 + 0.110i)15-s + 0.250·16-s − 0.329·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.420024 + 0.830990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.420024 + 0.830990i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.482 - 1.66i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-1.14 - 3.11i)T \) |
good | 5 | \( 1 - 0.885iT - 5T^{2} \) |
| 13 | \( 1 - 1.32iT - 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 4.67iT - 19T^{2} \) |
| 23 | \( 1 - 1.08iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 0.592T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 0.991iT - 43T^{2} \) |
| 47 | \( 1 - 0.204iT - 47T^{2} \) |
| 53 | \( 1 - 1.07iT - 53T^{2} \) |
| 59 | \( 1 - 5.25iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 - 6.28iT - 73T^{2} \) |
| 79 | \( 1 + 5.69iT - 79T^{2} \) |
| 83 | \( 1 - 9.95T + 83T^{2} \) |
| 89 | \( 1 + 6.72iT - 89T^{2} \) |
| 97 | \( 1 + 3.90T + 97T^{2} \) |
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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
−11.03194041327491720629177071476, −10.36002252595266070883291576065, −9.582086614233734539643811766005, −8.928408324437715649332789715020, −7.80170388854085740625336056560, −6.98542676794166871250947527857, −5.76581275761388937752616746592, −4.44636769518570400341956210641, −3.44931105707305201090059570924, −1.99907504728769682787969289511,
0.69245159263392142834254933445, 2.18604905905274320691309441904, 3.40300511044889341928815479493, 5.27991953916350640403676718743, 6.29698684847699958566126836870, 7.15842460192527492733608192299, 8.175926519190337480224346227005, 8.835421699087416991626632638180, 9.463311387949514137331940993913, 11.00029095967030828323809424735