L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.230 − 2.22i)5-s + (−0.432 + 0.749i)7-s + 0.999·8-s + (−1.81 + 1.31i)10-s + (−0.151 + 0.0874i)11-s + (−1.35 + 3.34i)13-s + 0.865·14-s + (−0.5 − 0.866i)16-s + (7.08 + 4.08i)17-s + (5.20 + 3.00i)19-s + (2.04 + 0.912i)20-s + (0.151 + 0.0874i)22-s + (2.52 − 1.45i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.103 − 0.994i)5-s + (−0.163 + 0.283i)7-s + 0.353·8-s + (−0.572 + 0.414i)10-s + (−0.0456 + 0.0263i)11-s + (−0.375 + 0.926i)13-s + 0.231·14-s + (−0.125 − 0.216i)16-s + (1.71 + 0.991i)17-s + (1.19 + 0.689i)19-s + (0.456 + 0.204i)20-s + (0.0322 + 0.0186i)22-s + (0.525 − 0.303i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.302233700\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.302233700\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.230 + 2.22i)T \) |
| 13 | \( 1 + (1.35 - 3.34i)T \) |
good | 7 | \( 1 + (0.432 - 0.749i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.151 - 0.0874i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-7.08 - 4.08i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.20 - 3.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 + 1.45i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.24 + 5.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.95iT - 31T^{2} \) |
| 37 | \( 1 + (-0.879 - 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.08 + 4.08i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.94 - 4.58i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 2.48iT - 53T^{2} \) |
| 59 | \( 1 + (-6.09 - 3.51i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.98 + 6.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.36 + 2.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.2 + 7.08i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 - 0.139T + 83T^{2} \) |
| 89 | \( 1 + (-11.3 + 6.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.32 + 7.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.504512003074996244489688236545, −9.199508227134224463178163111818, −7.88339275237030653442091486154, −7.76251336873891950334476714056, −6.15416836511072243630093699444, −5.36677905078872230714305225265, −4.30409616510728037624425523568, −3.46543472213599111382355000596, −2.08436633885495748729726412004, −0.959797651419678292825180786214,
0.919722955755061711518566904380, 2.83617440901365159023148120406, 3.50641203756179497083227463010, 5.11578776948308342825456985851, 5.61533091999786919462965133447, 6.88952927501411945079144022317, 7.35441979444720853597462968218, 7.935798235331537344126753088661, 9.185340349844614380060998849174, 9.845968185952774603085543921790