| L(s) = 1 | + 1.55·3-s − 3.85·5-s − 0.582·9-s + 11-s + 4.04·13-s − 5.98·15-s + 4.91·17-s − 3.80·19-s − 0.307·23-s + 9.82·25-s − 5.57·27-s − 8.45·29-s − 4.75·31-s + 1.55·33-s + 1.28·37-s + 6.29·39-s + 8.26·41-s − 2.43·43-s + 2.24·45-s + 10.9·47-s + 7.63·51-s − 7.43·53-s − 3.85·55-s − 5.91·57-s + 13.8·59-s − 8.81·61-s − 15.5·65-s + ⋯ |
| L(s) = 1 | + 0.897·3-s − 1.72·5-s − 0.194·9-s + 0.301·11-s + 1.12·13-s − 1.54·15-s + 1.19·17-s − 0.872·19-s − 0.0642·23-s + 1.96·25-s − 1.07·27-s − 1.57·29-s − 0.853·31-s + 0.270·33-s + 0.211·37-s + 1.00·39-s + 1.29·41-s − 0.371·43-s + 0.334·45-s + 1.59·47-s + 1.06·51-s − 1.02·53-s − 0.519·55-s − 0.783·57-s + 1.79·59-s − 1.12·61-s − 1.93·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699292328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699292328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 + 0.307T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.43T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 + 0.670T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 5.36T + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.290386958145870256408283053950, −7.76732970453714565204348627229, −7.29594219537714920157619909464, −6.22956895363035520616139629599, −5.42311719913170518390936467741, −4.21562410847489804102199882864, −3.69172296050915057814869103965, −3.26788614062304120638061497722, −2.06403932351571690760194084133, −0.69686605923867369712600940845,
0.69686605923867369712600940845, 2.06403932351571690760194084133, 3.26788614062304120638061497722, 3.69172296050915057814869103965, 4.21562410847489804102199882864, 5.42311719913170518390936467741, 6.22956895363035520616139629599, 7.29594219537714920157619909464, 7.76732970453714565204348627229, 8.290386958145870256408283053950
