L(s) = 1 | + (6.08 + 6.08i)5-s − 9.40i·7-s + (11.9 + 11.9i)11-s + (−3.47 − 3.47i)13-s + 28.5i·17-s + (−3.08 − 3.08i)19-s − 2.57·23-s + 49.1i·25-s + (−3.49 + 3.49i)29-s + 21.0·31-s + (57.2 − 57.2i)35-s + (13.2 − 13.2i)37-s − 11.2·41-s + (8.19 − 8.19i)43-s − 17.2i·47-s + ⋯ |
L(s) = 1 | + (1.21 + 1.21i)5-s − 1.34i·7-s + (1.08 + 1.08i)11-s + (−0.267 − 0.267i)13-s + 1.67i·17-s + (−0.162 − 0.162i)19-s − 0.111·23-s + 1.96i·25-s + (−0.120 + 0.120i)29-s + 0.680·31-s + (1.63 − 1.63i)35-s + (0.359 − 0.359i)37-s − 0.274·41-s + (0.190 − 0.190i)43-s − 0.366i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.535874733\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.535874733\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-6.08 - 6.08i)T + 25iT^{2} \) |
| 7 | \( 1 + 9.40iT - 49T^{2} \) |
| 11 | \( 1 + (-11.9 - 11.9i)T + 121iT^{2} \) |
| 13 | \( 1 + (3.47 + 3.47i)T + 169iT^{2} \) |
| 17 | \( 1 - 28.5iT - 289T^{2} \) |
| 19 | \( 1 + (3.08 + 3.08i)T + 361iT^{2} \) |
| 23 | \( 1 + 2.57T + 529T^{2} \) |
| 29 | \( 1 + (3.49 - 3.49i)T - 841iT^{2} \) |
| 31 | \( 1 - 21.0T + 961T^{2} \) |
| 37 | \( 1 + (-13.2 + 13.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 11.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-8.19 + 8.19i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 17.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-30.5 - 30.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-14.7 - 14.7i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (39.3 + 39.3i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-79.6 - 79.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 73.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 1.23iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (13.7 - 13.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 39.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−9.992422350428024062496699630370, −9.185553149723797174821572758589, −7.920021222111887472691684513234, −6.97229171634665759245818697509, −6.61704153231099186306044912721, −5.72581411638087883046355625686, −4.36303094298853863673285078967, −3.61456211182533019315879199980, −2.29842305326649906278421259352, −1.35662936267025751732363309927,
0.809644728972876514613631752513, 1.96637782808427751042862397834, 2.95649409149084414679082100325, 4.49711693994261196033263244258, 5.33616491143189575541466871953, 5.92262847715267855242036319970, 6.69461751985066614752755654770, 8.172705063595842532775373975123, 8.893119016972033637151734387344, 9.311898006010080844841070378778