| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8864958266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8864958266\) |
| \(L(1)\) |
\(\approx\) |
\(0.7089249512\) |
| \(L(1)\) |
\(\approx\) |
\(0.7089249512\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3253 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.74461179034300208353059006004, −18.48940093655265437679439499477, −17.86208350818024930094990975731, −16.4874321893603791014879995271, −15.99822576509820974410696933424, −15.68167532041767483983575200856, −15.110441204487524024166564399320, −14.06396957935222754508190862059, −13.20733558612506301678139822776, −12.66401816511276087918028810143, −11.76687372692600375265601259983, −10.99057203766866222486882155530, −10.26585345675856717394927164681, −9.59581878371254833156805230731, −8.89580573219067564512719212062, −8.12479199505338225540476499425, −7.84601875744310535676012014311, −6.85123084326625320587365823059, −6.40424988460877213063450447717, −5.10982116037203657328450129169, −3.90644650225220993597395194613, −3.30668249080491334801583262128, −2.701924894443123046844056639355, −1.729416400323669123045934121356, −0.56966390461506811342158310161,
0.56966390461506811342158310161, 1.729416400323669123045934121356, 2.701924894443123046844056639355, 3.30668249080491334801583262128, 3.90644650225220993597395194613, 5.10982116037203657328450129169, 6.40424988460877213063450447717, 6.85123084326625320587365823059, 7.84601875744310535676012014311, 8.12479199505338225540476499425, 8.89580573219067564512719212062, 9.59581878371254833156805230731, 10.26585345675856717394927164681, 10.99057203766866222486882155530, 11.76687372692600375265601259983, 12.66401816511276087918028810143, 13.20733558612506301678139822776, 14.06396957935222754508190862059, 15.110441204487524024166564399320, 15.68167532041767483983575200856, 15.99822576509820974410696933424, 16.4874321893603791014879995271, 17.86208350818024930094990975731, 18.48940093655265437679439499477, 18.74461179034300208353059006004
