| L(s) = 1 | − 1.60·2-s − 3-s + 0.565·4-s − 2.27·5-s + 1.60·6-s − 7-s + 2.29·8-s + 9-s + 3.64·10-s − 2.53·11-s − 0.565·12-s + 1.60·14-s + 2.27·15-s − 4.81·16-s − 4.05·17-s − 1.60·18-s − 7.93·19-s − 1.28·20-s + 21-s + 4.06·22-s + 0.458·23-s − 2.29·24-s + 0.188·25-s − 27-s − 0.565·28-s + 6.01·29-s − 3.64·30-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.282·4-s − 1.01·5-s + 0.653·6-s − 0.377·7-s + 0.812·8-s + 0.333·9-s + 1.15·10-s − 0.765·11-s − 0.163·12-s + 0.428·14-s + 0.588·15-s − 1.20·16-s − 0.983·17-s − 0.377·18-s − 1.82·19-s − 0.287·20-s + 0.218·21-s + 0.867·22-s + 0.0955·23-s − 0.469·24-s + 0.0377·25-s − 0.192·27-s − 0.106·28-s + 1.11·29-s − 0.666·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06008362711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06008362711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 23 | \( 1 - 0.458T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 - 0.340T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 4.25T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 + 9.42T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.426025922770439350320133967928, −8.124092834021294805873759525040, −7.08476354642695451406248521233, −6.73127586984559660402332705950, −5.63880165732207688749597837373, −4.49467343361494155726671406594, −4.22313817006679052745908847310, −2.84495953247700579946033400634, −1.67026932771570483956781515649, −0.17379618286753217049952657468,
0.17379618286753217049952657468, 1.67026932771570483956781515649, 2.84495953247700579946033400634, 4.22313817006679052745908847310, 4.49467343361494155726671406594, 5.63880165732207688749597837373, 6.73127586984559660402332705950, 7.08476354642695451406248521233, 8.124092834021294805873759525040, 8.426025922770439350320133967928
