| L(s) = 1 | − 2.55·2-s − 1.85·3-s + 4.54·4-s + 3.83·5-s + 4.73·6-s − 7-s − 6.51·8-s + 0.425·9-s − 9.80·10-s + 3.01·11-s − 8.41·12-s + 2.53·13-s + 2.55·14-s − 7.09·15-s + 7.56·16-s + 0.588·17-s − 1.08·18-s + 0.607·19-s + 17.4·20-s + 1.85·21-s − 7.70·22-s + 6.48·23-s + 12.0·24-s + 9.69·25-s − 6.47·26-s + 4.76·27-s − 4.54·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 1.06·3-s + 2.27·4-s + 1.71·5-s + 1.93·6-s − 0.377·7-s − 2.30·8-s + 0.141·9-s − 3.10·10-s + 0.908·11-s − 2.42·12-s + 0.701·13-s + 0.683·14-s − 1.83·15-s + 1.89·16-s + 0.142·17-s − 0.256·18-s + 0.139·19-s + 3.89·20-s + 0.403·21-s − 1.64·22-s + 1.35·23-s + 2.45·24-s + 1.93·25-s − 1.26·26-s + 0.917·27-s − 0.858·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9478768430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9478768430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 859 | \( 1 - T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 1.85T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 0.588T + 17T^{2} \) |
| 19 | \( 1 - 0.607T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 8.64T + 61T^{2} \) |
| 67 | \( 1 - 7.62T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 + 4.67T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.373961843327263042202104429076, −7.25042817092429976447604044579, −6.55715477505420240088380848641, −6.25275353744422412157925784311, −5.68504876861069090582114444596, −4.77383288620792973368828833039, −3.19936688637169145507977650656, −2.34665547309816992565766662310, −1.32759267722567847431740472839, −0.823478240830793306478428894032,
0.823478240830793306478428894032, 1.32759267722567847431740472839, 2.34665547309816992565766662310, 3.19936688637169145507977650656, 4.77383288620792973368828833039, 5.68504876861069090582114444596, 6.25275353744422412157925784311, 6.55715477505420240088380848641, 7.25042817092429976447604044579, 8.373961843327263042202104429076
