L(s) = 1 | + (1.25 + 0.660i)2-s + (1.12 + 1.65i)4-s − 1.68·7-s + (0.320 + 2.81i)8-s + 2.05i·11-s − 0.247·13-s + (−2.10 − 1.11i)14-s + (−1.45 + 3.72i)16-s + 1.13·17-s + 4.79·19-s + (−1.35 + 2.57i)22-s + 5.26i·23-s + (−0.309 − 0.163i)26-s + (−1.89 − 2.77i)28-s − 0.599·29-s + ⋯ |
L(s) = 1 | + (0.884 + 0.466i)2-s + (0.564 + 0.825i)4-s − 0.636·7-s + (0.113 + 0.993i)8-s + 0.619i·11-s − 0.0687·13-s + (−0.562 − 0.296i)14-s + (−0.363 + 0.931i)16-s + 0.275·17-s + 1.09·19-s + (−0.289 + 0.548i)22-s + 1.09i·23-s + (−0.0607 − 0.0320i)26-s + (−0.358 − 0.525i)28-s − 0.111·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.435900818\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.435900818\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.25 - 0.660i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.68T + 7T^{2} \) |
| 11 | \( 1 - 2.05iT - 11T^{2} \) |
| 13 | \( 1 + 0.247T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 - 5.26iT - 23T^{2} \) |
| 29 | \( 1 + 0.599T + 29T^{2} \) |
| 31 | \( 1 - 6.26iT - 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 3.83iT - 41T^{2} \) |
| 43 | \( 1 - 3.88iT - 43T^{2} \) |
| 47 | \( 1 + 6.96iT - 47T^{2} \) |
| 53 | \( 1 - 10.2iT - 53T^{2} \) |
| 59 | \( 1 + 0.397iT - 59T^{2} \) |
| 61 | \( 1 - 4.46iT - 61T^{2} \) |
| 67 | \( 1 + 0.232iT - 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 7.98iT - 73T^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 - 16.3iT - 89T^{2} \) |
| 97 | \( 1 + 16.4iT - 97T^{2} \) |
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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
−9.520252349564752876569846172054, −8.667981580907497597269663531514, −7.63695152710963790157158384604, −7.13597783623680076021367425718, −6.33449492243909519263818298662, −5.41892908000048825371011697207, −4.80544351352063838128558345327, −3.61056330303571871344473454367, −3.06325111773847047461544198920, −1.70240403749541893514156771089,
0.65185416424341848029295879579, 2.11344650230117582988621179267, 3.17864670473787561237346165505, 3.78093570983313104605708793844, 4.93148873102949129215965212452, 5.64803902944353009604170568310, 6.45895424609748066477027059759, 7.16022621179277639332552799157, 8.204456512872833333034181374104, 9.239909017875601206910196521919