L(s) = 1 | + 2-s + (1.37 − 1.05i)3-s + 4-s + 3.92i·5-s + (1.37 − 1.05i)6-s − i·7-s + 8-s + (0.775 − 2.89i)9-s + 3.92i·10-s + (−1.42 + 2.99i)11-s + (1.37 − 1.05i)12-s + 0.109i·13-s − i·14-s + (4.14 + 5.39i)15-s + 16-s + 6.97·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.793 − 0.608i)3-s + 0.5·4-s + 1.75i·5-s + (0.560 − 0.430i)6-s − 0.377i·7-s + 0.353·8-s + (0.258 − 0.965i)9-s + 1.24i·10-s + (−0.430 + 0.902i)11-s + (0.396 − 0.304i)12-s + 0.0302i·13-s − 0.267i·14-s + (1.06 + 1.39i)15-s + 0.250·16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.68162 + 0.281676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.68162 + 0.281676i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (1.42 - 2.99i)T \) |
good | 5 | \( 1 - 3.92iT - 5T^{2} \) |
| 13 | \( 1 - 0.109iT - 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 5.78iT - 19T^{2} \) |
| 23 | \( 1 - 0.938iT - 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 8.95T + 41T^{2} \) |
| 43 | \( 1 + 1.44iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 7.60iT - 59T^{2} \) |
| 61 | \( 1 - 6.35iT - 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 - 4.39iT - 71T^{2} \) |
| 73 | \( 1 + 3.31iT - 73T^{2} \) |
| 79 | \( 1 - 0.361iT - 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 0.722iT - 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−11.18239166078312965133208599926, −10.23443225459578875963507986711, −9.515742730473287814106010832036, −7.85421886962103693607627348888, −7.26144883462653383752410721469, −6.74103307118721843846100019470, −5.50217011549406991390891352990, −3.84004568112856240430263248887, −3.05870302742654619027027830745, −2.06548055137123558921920072923,
1.60908759030253174622253861528, 3.26039977561571755555978404721, 4.15922704546247809363293586548, 5.35513832329294414578413211118, 5.68887698141195406266729241251, 7.84949984276451969285555268439, 8.165981692227781631375777994506, 9.250260426497602149549453640197, 9.918981612949747025449075973948, 11.11889147242564190651454666779