| L(s) = 1 | + 5-s − 2.73·7-s − 5.97·11-s − 5.40·13-s − 6.94·17-s − 3.97·19-s + 23-s + 25-s + 9.29·29-s − 3.54·31-s − 2.73·35-s + 7.17·37-s − 10.1·41-s + 1.42·43-s − 10.4·47-s + 0.487·49-s + 4.04·53-s − 5.97·55-s + 8.13·59-s − 3.34·61-s − 5.40·65-s − 1.70·67-s − 7.56·71-s + 7.85·73-s + 16.3·77-s − 1.36·79-s + 13.9·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.03·7-s − 1.80·11-s − 1.49·13-s − 1.68·17-s − 0.912·19-s + 0.208·23-s + 0.200·25-s + 1.72·29-s − 0.636·31-s − 0.462·35-s + 1.18·37-s − 1.59·41-s + 0.217·43-s − 1.51·47-s + 0.0696·49-s + 0.556·53-s − 0.805·55-s + 1.05·59-s − 0.428·61-s − 0.669·65-s − 0.208·67-s − 0.897·71-s + 0.919·73-s + 1.86·77-s − 0.153·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5232175333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5232175333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 5.97T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 + 3.97T + 19T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 - 7.17T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 1.42T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.04T + 53T^{2} \) |
| 59 | \( 1 - 8.13T + 59T^{2} \) |
| 61 | \( 1 + 3.34T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 + 1.36T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.82602049399249134707841913227, −6.90202389469066873898691023300, −6.60407959943068444048070420442, −5.75174000354469733688254978686, −4.86578433825933726133727548798, −4.58033463405559768853368965771, −3.26102459286128709439405811516, −2.54422616734848551979751937159, −2.13272666985618256259963074159, −0.32240760495662175513553040452,
0.32240760495662175513553040452, 2.13272666985618256259963074159, 2.54422616734848551979751937159, 3.26102459286128709439405811516, 4.58033463405559768853368965771, 4.86578433825933726133727548798, 5.75174000354469733688254978686, 6.60407959943068444048070420442, 6.90202389469066873898691023300, 7.82602049399249134707841913227
