| L(s) = 1 | + 0.0888·2-s − 1.99·4-s + 0.674·5-s + 0.687·7-s − 0.354·8-s + 0.0599·10-s + 0.536·11-s − 0.628·13-s + 0.0611·14-s + 3.95·16-s − 5.05·17-s − 0.724·19-s − 1.34·20-s + 0.0477·22-s + 7.36·23-s − 4.54·25-s − 0.0558·26-s − 1.37·28-s + 3.61·29-s + 5.74·31-s + 1.06·32-s − 0.449·34-s + 0.463·35-s − 0.110·37-s − 0.0644·38-s − 0.239·40-s − 8.05·41-s + ⋯ |
| L(s) = 1 | + 0.0628·2-s − 0.996·4-s + 0.301·5-s + 0.260·7-s − 0.125·8-s + 0.0189·10-s + 0.161·11-s − 0.174·13-s + 0.0163·14-s + 0.988·16-s − 1.22·17-s − 0.166·19-s − 0.300·20-s + 0.0101·22-s + 1.53·23-s − 0.909·25-s − 0.0109·26-s − 0.258·28-s + 0.671·29-s + 1.03·31-s + 0.187·32-s − 0.0770·34-s + 0.0784·35-s − 0.0182·37-s − 0.0104·38-s − 0.0378·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435458817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435458817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0888T + 2T^{2} \) |
| 5 | \( 1 - 0.674T + 5T^{2} \) |
| 7 | \( 1 - 0.687T + 7T^{2} \) |
| 11 | \( 1 - 0.536T + 11T^{2} \) |
| 13 | \( 1 + 0.628T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.724T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 + 0.110T + 37T^{2} \) |
| 41 | \( 1 + 8.05T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 9.81T + 59T^{2} \) |
| 61 | \( 1 - 6.30T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−8.895247987715656915826930940702, −8.605832345099405287734300675809, −7.59396934191499772631941251495, −6.69530165544076608208974425068, −5.84850950681180972265743930560, −4.89840613528370836121914020695, −4.40919668339774802539130757061, −3.34619109400038917775928488387, −2.19147804479805430018169871320, −0.792898479980978233185509339486,
0.792898479980978233185509339486, 2.19147804479805430018169871320, 3.34619109400038917775928488387, 4.40919668339774802539130757061, 4.89840613528370836121914020695, 5.84850950681180972265743930560, 6.69530165544076608208974425068, 7.59396934191499772631941251495, 8.605832345099405287734300675809, 8.895247987715656915826930940702
