L(s) = 1 | − 5-s + 4.21·7-s + 11-s + 5.57·13-s + 2.21·17-s − 0.643·19-s − 0.643·23-s + 25-s − 2·29-s + 1.35·31-s − 4.21·35-s + 2·37-s − 2·41-s + 4.86·43-s + 4.64·47-s + 10.7·49-s + 6.43·53-s − 55-s + 1.35·59-s − 0.0701·61-s − 5.57·65-s + 12.4·67-s − 5.14·71-s + 10.6·73-s + 4.21·77-s − 7.14·79-s + 13.9·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.59·7-s + 0.301·11-s + 1.54·13-s + 0.537·17-s − 0.147·19-s − 0.134·23-s + 0.200·25-s − 0.371·29-s + 0.243·31-s − 0.713·35-s + 0.328·37-s − 0.312·41-s + 0.741·43-s + 0.677·47-s + 1.54·49-s + 0.884·53-s − 0.134·55-s + 0.176·59-s − 0.00898·61-s − 0.691·65-s + 1.51·67-s − 0.611·71-s + 1.24·73-s + 0.480·77-s − 0.804·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.821887581\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.821887581\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 4.21T + 7T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 2.21T + 17T^{2} \) |
| 19 | \( 1 + 0.643T + 19T^{2} \) |
| 23 | \( 1 + 0.643T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 + 0.0701T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 7.72T + 97T^{2} \) |
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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
−8.000562410581236176874584018304, −7.27877872269920851009012261268, −6.46952288873415835745020589509, −5.65182422483677650554724055030, −5.09463510357571939565624763040, −4.09962227094541030688644284897, −3.82555965982813357286118015104, −2.62448910976994393767144653487, −1.59549443629513187607759390112, −0.927976871389569294018248088191,
0.927976871389569294018248088191, 1.59549443629513187607759390112, 2.62448910976994393767144653487, 3.82555965982813357286118015104, 4.09962227094541030688644284897, 5.09463510357571939565624763040, 5.65182422483677650554724055030, 6.46952288873415835745020589509, 7.27877872269920851009012261268, 8.000562410581236176874584018304