| L(s) = 1 | + 0.732·5-s + 2·7-s + 1.46·11-s + 3.46·13-s − 0.732·17-s + 5.46·19-s − 1.26·23-s − 4.46·25-s + 6.19·29-s + 4·31-s + 1.46·35-s − 37-s + 2·41-s + 10.9·43-s − 6.92·47-s − 3·49-s − 10.3·53-s + 1.07·55-s + 1.26·59-s + 4.92·61-s + 2.53·65-s + 6·67-s + 4·71-s − 5.46·73-s + 2.92·77-s + 2.53·79-s − 2.53·83-s + ⋯ |
| L(s) = 1 | + 0.327·5-s + 0.755·7-s + 0.441·11-s + 0.960·13-s − 0.177·17-s + 1.25·19-s − 0.264·23-s − 0.892·25-s + 1.15·29-s + 0.718·31-s + 0.247·35-s − 0.164·37-s + 0.312·41-s + 1.66·43-s − 1.01·47-s − 0.428·49-s − 1.42·53-s + 0.144·55-s + 0.165·59-s + 0.630·61-s + 0.314·65-s + 0.733·67-s + 0.474·71-s − 0.639·73-s + 0.333·77-s + 0.285·79-s − 0.278·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.712797826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712797826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 - 2.19T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−8.140219467958143228184256331553, −7.62827345347524981577459047127, −6.63511945972764432231670129640, −6.07304472433564017469419878128, −5.29678599923016106034016525037, −4.53086224868694803075884730582, −3.74345415969305792894469203657, −2.82644360521459512449818109416, −1.75252324282570182470191304846, −0.963929002291009989050359340552,
0.963929002291009989050359340552, 1.75252324282570182470191304846, 2.82644360521459512449818109416, 3.74345415969305792894469203657, 4.53086224868694803075884730582, 5.29678599923016106034016525037, 6.07304472433564017469419878128, 6.63511945972764432231670129640, 7.62827345347524981577459047127, 8.140219467958143228184256331553
