L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.361 + 1.11i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (−0.947 − 0.688i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (1.31 − 0.956i)9-s + 10-s + (−3.15 + 1.02i)11-s + 1.17·12-s + (−2.19 + 1.59i)13-s + (0.309 + 0.951i)14-s + (0.361 − 1.11i)15-s + (−0.809 − 0.587i)16-s + (4.32 + 3.14i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.208 + 0.643i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.386 − 0.281i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (0.438 − 0.318i)9-s + 0.316·10-s + (−0.951 + 0.308i)11-s + 0.338·12-s + (−0.609 + 0.442i)13-s + (0.0825 + 0.254i)14-s + (0.0934 − 0.287i)15-s + (−0.202 − 0.146i)16-s + (1.04 + 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.791530 + 0.772231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.791530 + 0.772231i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (3.15 - 1.02i)T \) |
good | 3 | \( 1 + (-0.361 - 1.11i)T + (-2.42 + 1.76i)T^{2} \) |
| 13 | \( 1 + (2.19 - 1.59i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.32 - 3.14i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 3.78i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 5.49T + 23T^{2} \) |
| 29 | \( 1 + (1.54 - 4.75i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 1.68i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.0125 + 0.0385i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.58 - 4.87i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + (-3.05 - 9.41i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.68 - 2.67i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.180 - 0.556i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.87 + 4.26i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + (6.64 + 4.82i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.50 + 13.8i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.46 - 3.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-12.6 - 9.15i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + (-1.07 + 0.780i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.40343672259948340660211805339, −9.590729083380565205517227693647, −8.981897203109293179749758279704, −7.73218972063392278875870851518, −7.50434742296780965280609221150, −6.19333199179498137159439553831, −5.07150725510958823617571675352, −4.27593222279663805073270573898, −3.07456264673264895183269653848, −1.30341256162744403269131759346,
0.75643666298389778636423521083, 2.40349486086119960824061949296, 3.05623057799027715855863817503, 4.65688489794623829101061017285, 5.65900299765857466550038953934, 7.20116788938384723861108151113, 7.45206898897656156305557650563, 8.307583733362007115489423210663, 9.254076857395620030452191975275, 10.19178822431862183883843830321