| L(s) = 1 | − 3-s + 1.23·5-s + 5.23·7-s + 9-s + 4.47·11-s − 13-s − 1.23·15-s + 4.47·17-s − 1.23·19-s − 5.23·21-s − 2.47·23-s − 3.47·25-s − 27-s − 8.47·29-s + 9.23·31-s − 4.47·33-s + 6.47·35-s − 4.47·37-s + 39-s − 9.23·41-s + 6.47·43-s + 1.23·45-s − 0.472·47-s + 20.4·49-s − 4.47·51-s + 0.472·53-s + 5.52·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.552·5-s + 1.97·7-s + 0.333·9-s + 1.34·11-s − 0.277·13-s − 0.319·15-s + 1.08·17-s − 0.283·19-s − 1.14·21-s − 0.515·23-s − 0.694·25-s − 0.192·27-s − 1.57·29-s + 1.65·31-s − 0.778·33-s + 1.09·35-s − 0.735·37-s + 0.160·39-s − 1.44·41-s + 0.986·43-s + 0.184·45-s − 0.0688·47-s + 2.91·49-s − 0.626·51-s + 0.0648·53-s + 0.745·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050655626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050655626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 6.94T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−9.801953882476658130959436027994, −8.874472604324464492396088391494, −8.021488268728704913289073167464, −7.30930144294875900153204955992, −6.21249569375365825074518450026, −5.47549923340203215077026174519, −4.68390701152853622123710499884, −3.80662275402744321223719606504, −2.00782760599828459918974298646, −1.27538981329969259346157492479,
1.27538981329969259346157492479, 2.00782760599828459918974298646, 3.80662275402744321223719606504, 4.68390701152853622123710499884, 5.47549923340203215077026174519, 6.21249569375365825074518450026, 7.30930144294875900153204955992, 8.021488268728704913289073167464, 8.874472604324464492396088391494, 9.801953882476658130959436027994
