L(s) = 1 | + (−1.29 − 0.576i)2-s + (1.33 + 1.48i)4-s + 1.97·7-s + (−0.867 − 2.69i)8-s + 1.43i·11-s − 0.241i·13-s + (−2.55 − 1.13i)14-s + (−0.430 + 3.97i)16-s − 7.38·17-s + 3.04i·19-s + (0.824 − 1.84i)22-s + 0.874·23-s + (−0.139 + 0.311i)26-s + (2.64 + 2.94i)28-s + 9.07i·29-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.407i)2-s + (0.667 + 0.744i)4-s + 0.747·7-s + (−0.306 − 0.951i)8-s + 0.431i·11-s − 0.0669i·13-s + (−0.682 − 0.304i)14-s + (−0.107 + 0.994i)16-s − 1.79·17-s + 0.697i·19-s + (0.175 − 0.393i)22-s + 0.182·23-s + (−0.0272 + 0.0611i)26-s + (0.499 + 0.556i)28-s + 1.68i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5776460201\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5776460201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 + 0.576i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.241iT - 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 - 3.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.874T + 23T^{2} \) |
| 29 | \( 1 - 9.07iT - 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 8.81iT - 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 + 9.20iT - 53T^{2} \) |
| 59 | \( 1 - 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 - 4.86iT - 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 - 4.12T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.443422019105097488652786898252, −8.677769433537877650072379605029, −8.225820941178381934293535052476, −7.13896863568168104920598622440, −6.75661406613353980431088948892, −5.45562516587524674874706891489, −4.45821108776312668399457771628, −3.48513000438817746203167674559, −2.26059746245436191671724008457, −1.46999755620462145334551293533,
0.28077978081483057188007821664, 1.76196450173472859397856046986, 2.64602463544845509111567834608, 4.21164284708796647128687880914, 5.07442722198735723964743801859, 6.05313912855157517836348627974, 6.73237211190042692745916107320, 7.64280113003288771967584465071, 8.218600582537886483142135343905, 9.118926933514869006459768719397