| L(s) = 1 | + (1.39 − 0.205i)2-s + 3.38i·3-s + (1.91 − 0.575i)4-s + (0.695 + 4.73i)6-s − 1.47i·7-s + (2.56 − 1.19i)8-s − 8.45·9-s + 5.75i·11-s + (1.94 + 6.48i)12-s + (−0.303 − 2.06i)14-s + (3.33 − 2.20i)16-s + (−11.8 + 1.73i)18-s + 7.28i·19-s + 4.99·21-s + (1.18 + 8.05i)22-s + ⋯ |
| L(s) = 1 | + (0.989 − 0.145i)2-s + 1.95i·3-s + (0.957 − 0.287i)4-s + (0.283 + 1.93i)6-s − 0.557i·7-s + (0.905 − 0.423i)8-s − 2.81·9-s + 1.73i·11-s + (0.561 + 1.87i)12-s + (−0.0810 − 0.551i)14-s + (0.834 − 0.550i)16-s + (−2.78 + 0.409i)18-s + 1.67i·19-s + 1.09·21-s + (0.252 + 1.71i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56410 + 2.10258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56410 + 2.10258i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.205i)T \) |
| 167 | \( 1 - 12.9iT \) |
| good | 3 | \( 1 - 3.38iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 1.47iT - 7T^{2} \) |
| 11 | \( 1 - 5.75iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 7.28iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 + 9.96iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−10.60359823746337628820656128868, −10.06867162440355646160591912922, −9.572555199560182297261754668024, −8.205347898833459806766474168208, −7.08383287084267301636332039756, −5.82542991718399765634279216805, −5.02670299930669102675973000972, −4.16866903918201451533462032604, −3.71531763944358833464009536989, −2.31473564798838392458910518412,
1.07754219846462816133299238470, 2.56547624958928354818923643920, 3.16278108800227762196426616832, 5.13430942568951181565001222417, 5.89058027932566407844360916924, 6.65494707514159863705337255445, 7.27539640800568508358511481188, 8.407280701719127561330146252682, 8.799601898874515469756800947618, 11.04824673641361708899300376087
