L(s) = 1 | + 0.628·2-s + (−1.53 − 0.792i)3-s − 1.60·4-s + 1.39i·5-s + (−0.968 − 0.498i)6-s + i·7-s − 2.26·8-s + (1.74 + 2.44i)9-s + 0.880i·10-s + (−1.97 + 2.66i)11-s + (2.47 + 1.27i)12-s + 5.12i·13-s + 0.628i·14-s + (1.10 − 2.15i)15-s + 1.78·16-s − 7.86·17-s + ⋯ |
L(s) = 1 | + 0.444·2-s + (−0.889 − 0.457i)3-s − 0.802·4-s + 0.625i·5-s + (−0.395 − 0.203i)6-s + 0.377i·7-s − 0.801·8-s + (0.580 + 0.814i)9-s + 0.278i·10-s + (−0.596 + 0.802i)11-s + (0.713 + 0.367i)12-s + 1.42i·13-s + 0.168i·14-s + (0.286 − 0.556i)15-s + 0.445·16-s − 1.90·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.285812 + 0.458277i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.285812 + 0.458277i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.53 + 0.792i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.97 - 2.66i)T \) |
good | 2 | \( 1 - 0.628T + 2T^{2} \) |
| 5 | \( 1 - 1.39iT - 5T^{2} \) |
| 13 | \( 1 - 5.12iT - 13T^{2} \) |
| 17 | \( 1 + 7.86T + 17T^{2} \) |
| 19 | \( 1 + 5.38iT - 19T^{2} \) |
| 23 | \( 1 - 2.26iT - 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 4.29iT - 43T^{2} \) |
| 47 | \( 1 - 6.76iT - 47T^{2} \) |
| 53 | \( 1 - 7.79iT - 53T^{2} \) |
| 59 | \( 1 + 14.8iT - 59T^{2} \) |
| 61 | \( 1 - 9.28iT - 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 - 9.34iT - 71T^{2} \) |
| 73 | \( 1 - 9.47iT - 73T^{2} \) |
| 79 | \( 1 - 2.26iT - 79T^{2} \) |
| 83 | \( 1 - 5.64T + 83T^{2} \) |
| 89 | \( 1 - 3.80iT - 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−12.61991797495052526422815434755, −11.58471950490205922094961757696, −10.88343839281991314105980741609, −9.617512132170461284211399069528, −8.686872610579259328584077754278, −7.10651779730361083568186864171, −6.47456154729499834537530847469, −5.08343793029092349203438326539, −4.36202709005412141551738179657, −2.34904654367329343488213549637,
0.42732592584369035689722484920, 3.48238889792221431320277531436, 4.65200728491639398289207418718, 5.38795762865740327041245998394, 6.38929313835152803648192418781, 8.126914341496560741507297327088, 8.954603083537243576162994011446, 10.18047058480266060095359588689, 10.82920598435027970170336760921, 12.05727554065111916072609156652