L(s) = 1 | + (0.321 + 1.37i)2-s + (−1.79 + 0.884i)4-s + (−2.11 − 0.726i)5-s − 4.05i·7-s + (−1.79 − 2.18i)8-s + (0.321 − 3.14i)10-s − 0.985i·11-s − 4.94·13-s + (5.58 − 1.30i)14-s + (2.43 − 3.17i)16-s − 4.52i·17-s + 2.60i·19-s + (4.43 − 0.567i)20-s + (1.35 − 0.316i)22-s + 3.53i·23-s + ⋯ |
L(s) = 1 | + (0.227 + 0.973i)2-s + (−0.896 + 0.442i)4-s + (−0.945 − 0.324i)5-s − 1.53i·7-s + (−0.634 − 0.773i)8-s + (0.101 − 0.994i)10-s − 0.297i·11-s − 1.37·13-s + (1.49 − 0.348i)14-s + (0.608 − 0.793i)16-s − 1.09i·17-s + 0.597i·19-s + (0.991 − 0.126i)20-s + (0.289 − 0.0674i)22-s + 0.737i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.515508 - 0.358186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.515508 - 0.358186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.321 - 1.37i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.11 + 0.726i)T \) |
good | 7 | \( 1 + 4.05iT - 7T^{2} \) |
| 11 | \( 1 + 0.985iT - 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 + 4.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 - 3.53iT - 23T^{2} \) |
| 29 | \( 1 + 7.59iT - 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 - 0.945T + 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 2.60iT - 47T^{2} \) |
| 53 | \( 1 - 0.229T + 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.23iT - 97T^{2} \) |
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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
−11.45136069876150635318286011323, −10.14136227678360720112744930046, −9.341216376434225342517312728364, −7.982589326297066977931987923622, −7.52802702286445053638182974961, −6.76723287610848661273650751553, −5.21524936965601010373350502873, −4.36347047387001100555464289813, −3.43687726664174114817933444810, −0.39259526158820347161299115304,
2.19010116685950845437714571762, 3.19112753406882920780044722043, 4.54964378579797929068876697320, 5.44105061759257269436399407651, 6.82189816448155959118084237339, 8.207865890739164026068846006429, 8.912812884343134730364818024993, 9.916598911018470403072561076241, 10.90048234517454770658564513788, 11.71613133451437851252561251168