L(s) = 1 | + 1.78·2-s + (−1.68 + 0.395i)3-s + 1.17·4-s − i·5-s + (−3.00 + 0.704i)6-s + i·7-s − 1.47·8-s + (2.68 − 1.33i)9-s − 1.78i·10-s + (−1.85 − 2.74i)11-s + (−1.98 + 0.464i)12-s + 5.68i·13-s + 1.78i·14-s + (0.395 + 1.68i)15-s − 4.96·16-s − 2.05·17-s + ⋯ |
L(s) = 1 | + 1.25·2-s + (−0.973 + 0.228i)3-s + 0.587·4-s − 0.447i·5-s + (−1.22 + 0.287i)6-s + 0.377i·7-s − 0.519·8-s + (0.895 − 0.444i)9-s − 0.563i·10-s + (−0.560 − 0.828i)11-s + (−0.571 + 0.134i)12-s + 1.57i·13-s + 0.476i·14-s + (0.102 + 0.435i)15-s − 1.24·16-s − 0.497·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8622667794\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8622667794\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.68 - 0.395i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.85 + 2.74i)T \) |
good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 13 | \( 1 - 5.68iT - 13T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 - 4.08iT - 19T^{2} \) |
| 23 | \( 1 - 5.58iT - 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 - 1.16iT - 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 4.49iT - 53T^{2} \) |
| 59 | \( 1 - 3.29iT - 59T^{2} \) |
| 61 | \( 1 - 2.93iT - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 7.13iT - 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 + 8.84iT - 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 + 0.807iT - 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−10.25547650970883859898498261553, −9.296101090625529107065410556180, −8.632155085087560134207164102824, −7.28241752672931087361154730559, −6.29317422690743601629391890256, −5.73556653373060964214347790873, −4.99011132979482646573235330391, −4.22539930015025477991904945211, −3.37794348202057981505056587032, −1.77880446640342273337178313341,
0.26425062641820782683152429747, 2.30104304428344704863588367708, 3.37116339970512896089099049260, 4.58384419591981588739505607884, 5.04467242374030564554553480769, 5.94689707739724519688235177648, 6.75238611037492305495036493795, 7.40597724973535253546589021966, 8.512335379013284532906764035952, 9.893718489898157917396663223854