| L(s) = 1 | + 2.77·2-s + 5.67·4-s + 2.13·5-s − 3.92·7-s + 10.1·8-s + 5.92·10-s − 0.269·11-s − 5.92·13-s − 10.8·14-s + 16.8·16-s − 7.41·17-s + 2·19-s + 12.1·20-s − 0.747·22-s + 0.269·23-s − 0.422·25-s − 16.4·26-s − 22.2·28-s + 3.13·29-s + 4.50·31-s + 26.3·32-s − 20.5·34-s − 8.40·35-s + 6·37-s + 5.54·38-s + 21.7·40-s + 1.05·41-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.83·4-s + 0.956·5-s − 1.48·7-s + 3.59·8-s + 1.87·10-s − 0.0813·11-s − 1.64·13-s − 2.90·14-s + 4.21·16-s − 1.79·17-s + 0.458·19-s + 2.71·20-s − 0.159·22-s + 0.0562·23-s − 0.0845·25-s − 3.22·26-s − 4.21·28-s + 0.581·29-s + 0.809·31-s + 4.65·32-s − 3.52·34-s − 1.42·35-s + 0.986·37-s + 0.898·38-s + 3.44·40-s + 0.165·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.509792645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.509792645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 11 | \( 1 + 0.269T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 7.41T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 0.269T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 - 4.81T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 4.81T + 83T^{2} \) |
| 89 | \( 1 - 5.79T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−11.01986992195206281003724783467, −10.07744942807200951668403743836, −9.410211989004412012011162914684, −7.54165679961406663629180712634, −6.59565152425129466251971329006, −6.19122010767529403444099262642, −5.11655011286013937734885156030, −4.25427615030360921636752815688, −2.88626513497237303181102357605, −2.30479490588331288532918105893,
2.30479490588331288532918105893, 2.88626513497237303181102357605, 4.25427615030360921636752815688, 5.11655011286013937734885156030, 6.19122010767529403444099262642, 6.59565152425129466251971329006, 7.54165679961406663629180712634, 9.410211989004412012011162914684, 10.07744942807200951668403743836, 11.01986992195206281003724783467
