| L(s) = 1 | − 1.83·5-s − 1.39·7-s − 5.50·11-s + 3.03·13-s + 3.51·17-s + 19-s − 7.65·23-s − 1.63·25-s − 5.01·29-s + 1.12·31-s + 2.55·35-s + 2.12·37-s + 1.11·41-s + 6.57·43-s − 8.55·47-s − 5.06·49-s + 4.63·53-s + 10.0·55-s + 4.38·59-s − 12.9·61-s − 5.56·65-s − 10.9·67-s − 6.63·71-s + 5.02·73-s + 7.65·77-s + 12.8·79-s − 4.40·83-s + ⋯ |
| L(s) = 1 | − 0.820·5-s − 0.525·7-s − 1.65·11-s + 0.841·13-s + 0.852·17-s + 0.229·19-s − 1.59·23-s − 0.326·25-s − 0.931·29-s + 0.202·31-s + 0.431·35-s + 0.349·37-s + 0.173·41-s + 1.00·43-s − 1.24·47-s − 0.723·49-s + 0.636·53-s + 1.36·55-s + 0.570·59-s − 1.65·61-s − 0.690·65-s − 1.33·67-s − 0.787·71-s + 0.588·73-s + 0.871·77-s + 1.45·79-s − 0.483·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8099203285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8099203285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 4.63T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 - 1.28T + 89T^{2} \) |
| 97 | \( 1 - 7.94T + 97T^{2} \) |
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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
−7.74213979829263800434141597611, −7.47525327521491655237279976663, −6.26253966121988600426162035426, −5.83477148832377417345459694404, −5.05346851858223730578132282989, −4.14354478220545458271324089398, −3.50425470198925400160253687944, −2.82836830458771698439861381843, −1.79100538487346869795093459953, −0.42773913653043163237175045575,
0.42773913653043163237175045575, 1.79100538487346869795093459953, 2.82836830458771698439861381843, 3.50425470198925400160253687944, 4.14354478220545458271324089398, 5.05346851858223730578132282989, 5.83477148832377417345459694404, 6.26253966121988600426162035426, 7.47525327521491655237279976663, 7.74213979829263800434141597611
