| L(s) = 1 | + 1.16·3-s + 4.97·7-s − 1.63·9-s + 11-s + 0.665·13-s − 6.77·17-s − 19-s + 5.80·21-s + 2.16·23-s − 5.41·27-s + 7.97·29-s + 8.94·31-s + 1.16·33-s + 0.139·37-s + 0.776·39-s − 1.80·41-s + 2.80·43-s + 0.530·47-s + 17.7·49-s − 7.91·51-s + 6.30·53-s − 1.16·57-s − 11.4·59-s + 10.5·61-s − 8.13·63-s + 9.60·67-s + 2.53·69-s + ⋯ |
| L(s) = 1 | + 0.674·3-s + 1.87·7-s − 0.545·9-s + 0.301·11-s + 0.184·13-s − 1.64·17-s − 0.229·19-s + 1.26·21-s + 0.451·23-s − 1.04·27-s + 1.48·29-s + 1.60·31-s + 0.203·33-s + 0.0229·37-s + 0.124·39-s − 0.281·41-s + 0.427·43-s + 0.0773·47-s + 2.53·49-s − 1.10·51-s + 0.865·53-s − 0.154·57-s − 1.49·59-s + 1.35·61-s − 1.02·63-s + 1.17·67-s + 0.304·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.042092438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.042092438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.16T + 3T^{2} \) |
| 7 | \( 1 - 4.97T + 7T^{2} \) |
| 13 | \( 1 - 0.665T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 0.139T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 - 0.530T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.393947866349123587138921320636, −7.961245865420031853913978591593, −6.95385519584931595159899740785, −6.26327823046298245974720299930, −5.18784479365775872555368119522, −4.62085997033475038002901758821, −3.91802467582255600375743306679, −2.66058598214247115968670586950, −2.11790545417944915247712233306, −0.990887517253212204352614077777,
0.990887517253212204352614077777, 2.11790545417944915247712233306, 2.66058598214247115968670586950, 3.91802467582255600375743306679, 4.62085997033475038002901758821, 5.18784479365775872555368119522, 6.26327823046298245974720299930, 6.95385519584931595159899740785, 7.961245865420031853913978591593, 8.393947866349123587138921320636
