L(s) = 1 | + 3.67·5-s + 4.79·7-s + 1.91·11-s + 3.05·13-s − 2.92·17-s − 8.17·19-s + 6.97·23-s + 8.49·25-s − 3.87·29-s + 0.104·31-s + 17.6·35-s − 1.37·37-s − 10.4·41-s − 5.08·43-s + 10.8·47-s + 16.0·49-s + 7.42·53-s + 7.03·55-s + 0.535·59-s + 8.80·61-s + 11.2·65-s − 8.50·67-s − 5.00·71-s + 8.80·73-s + 9.18·77-s − 6.76·79-s + 10.3·83-s + ⋯ |
L(s) = 1 | + 1.64·5-s + 1.81·7-s + 0.577·11-s + 0.847·13-s − 0.710·17-s − 1.87·19-s + 1.45·23-s + 1.69·25-s − 0.720·29-s + 0.0187·31-s + 2.97·35-s − 0.226·37-s − 1.63·41-s − 0.774·43-s + 1.57·47-s + 2.28·49-s + 1.01·53-s + 0.948·55-s + 0.0696·59-s + 1.12·61-s + 1.39·65-s − 1.03·67-s − 0.594·71-s + 1.03·73-s + 1.04·77-s − 0.761·79-s + 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.857888252\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.857888252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 0.104T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 - 0.535T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 8.50T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.421762325437819772174051103110, −7.22765687842974332181022238081, −6.59521794744278910396088553871, −5.92354562467733104465354368046, −5.19575332393618507876854970648, −4.63896905858780008601184737755, −3.76377947829743053550678047991, −2.36524230388126780159703868212, −1.87146512094186045157582716350, −1.15799914776327920911314078862,
1.15799914776327920911314078862, 1.87146512094186045157582716350, 2.36524230388126780159703868212, 3.76377947829743053550678047991, 4.63896905858780008601184737755, 5.19575332393618507876854970648, 5.92354562467733104465354368046, 6.59521794744278910396088553871, 7.22765687842974332181022238081, 8.421762325437819772174051103110