| L(s) = 1 | − 2-s + 0.0101·3-s + 4-s − 3.08·5-s − 0.0101·6-s + 7-s − 8-s − 2.99·9-s + 3.08·10-s − 6.48·11-s + 0.0101·12-s − 1.04·13-s − 14-s − 0.0312·15-s + 16-s + 2.51·17-s + 2.99·18-s − 3.08·20-s + 0.0101·21-s + 6.48·22-s − 8.22·23-s − 0.0101·24-s + 4.54·25-s + 1.04·26-s − 0.0606·27-s + 28-s − 5.21·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.00583·3-s + 0.5·4-s − 1.38·5-s − 0.00412·6-s + 0.377·7-s − 0.353·8-s − 0.999·9-s + 0.977·10-s − 1.95·11-s + 0.00291·12-s − 0.288·13-s − 0.267·14-s − 0.00805·15-s + 0.250·16-s + 0.609·17-s + 0.707·18-s − 0.690·20-s + 0.00220·21-s + 1.38·22-s − 1.71·23-s − 0.00206·24-s + 0.909·25-s + 0.203·26-s − 0.0116·27-s + 0.188·28-s − 0.969·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08099109628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08099109628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.0101T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 8.22T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 - 0.538T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 + 2.05T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.033303014809454018368486431518, −7.70515099369081531926403341729, −7.32855518524736977047541784372, −5.92918374143476855217395089180, −5.48123251336065139608714664398, −4.53170407423311943102012689421, −3.54947493083401360401554006257, −2.83563192843835080002915014729, −1.89635927742339144773713392927, −0.16062235867855174159972811855,
0.16062235867855174159972811855, 1.89635927742339144773713392927, 2.83563192843835080002915014729, 3.54947493083401360401554006257, 4.53170407423311943102012689421, 5.48123251336065139608714664398, 5.92918374143476855217395089180, 7.32855518524736977047541784372, 7.70515099369081531926403341729, 8.033303014809454018368486431518
