L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (1.24 + 1.70i)5-s + (0.951 − 0.309i)7-s + (0.309 − 0.951i)8-s − 2.11i·10-s + (−0.422 + 3.28i)11-s + (−0.352 + 0.485i)13-s + (−0.951 − 0.309i)14-s + (−0.809 + 0.587i)16-s + (0.694 − 0.504i)17-s + (2.22 + 0.721i)19-s + (−1.24 + 1.70i)20-s + (2.27 − 2.41i)22-s + 2.92i·23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.555 + 0.764i)5-s + (0.359 − 0.116i)7-s + (0.109 − 0.336i)8-s − 0.668i·10-s + (−0.127 + 0.991i)11-s + (−0.0978 + 0.134i)13-s + (−0.254 − 0.0825i)14-s + (−0.202 + 0.146i)16-s + (0.168 − 0.122i)17-s + (0.509 + 0.165i)19-s + (−0.277 + 0.382i)20-s + (0.485 − 0.514i)22-s + 0.610i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.309556820\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.309556820\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (0.422 - 3.28i)T \) |
good | 5 | \( 1 + (-1.24 - 1.70i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (0.352 - 0.485i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.694 + 0.504i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.22 - 0.721i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 2.92iT - 23T^{2} \) |
| 29 | \( 1 + (-1.88 - 5.81i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.894 - 0.649i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.37 + 7.31i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.44 - 10.6i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.69iT - 43T^{2} \) |
| 47 | \( 1 + (5.90 + 1.91i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.49 - 2.05i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.201 + 0.0653i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-7.31 - 10.0i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 2.58T + 67T^{2} \) |
| 71 | \( 1 + (-2.88 - 3.96i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.80 + 0.585i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.03 - 6.92i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (6.58 - 4.78i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 + (-5.61 - 4.07i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.919153710730956825420928111413, −9.073446207310954075983793221428, −8.128692701055414319911615402598, −7.25466524420887197095454327229, −6.74616475549317518032394370681, −5.57152214387950026855671589519, −4.59250427700207278423563373972, −3.39260168498466901640915456080, −2.42194329566808413465494040792, −1.43865850183962914799611196843,
0.67787328852214306553424419515, 1.86816090825829420211914789431, 3.19115318756308004098943657052, 4.63186358001829918455464111802, 5.40306810396394324345362141345, 6.08770373090590610842994605798, 7.03060240589334226361340563608, 8.173729605528136129977183556651, 8.459165040061189211404041910813, 9.378205687422738796829621178575