| L(s) = 1 | − 2.32·2-s + 0.535·3-s + 3.39·4-s − 0.511·5-s − 1.24·6-s + 3.83·7-s − 3.23·8-s − 2.71·9-s + 1.18·10-s − 11-s + 1.81·12-s + 1.58·13-s − 8.90·14-s − 0.273·15-s + 0.721·16-s + 5.58·17-s + 6.30·18-s + 6.53·19-s − 1.73·20-s + 2.05·21-s + 2.32·22-s − 3.91·23-s − 1.72·24-s − 4.73·25-s − 3.68·26-s − 3.05·27-s + 13.0·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 0.308·3-s + 1.69·4-s − 0.228·5-s − 0.507·6-s + 1.44·7-s − 1.14·8-s − 0.904·9-s + 0.375·10-s − 0.301·11-s + 0.523·12-s + 0.439·13-s − 2.37·14-s − 0.0706·15-s + 0.180·16-s + 1.35·17-s + 1.48·18-s + 1.49·19-s − 0.387·20-s + 0.447·21-s + 0.495·22-s − 0.816·23-s − 0.353·24-s − 0.947·25-s − 0.722·26-s − 0.588·27-s + 2.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124818501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124818501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 0.535T + 3T^{2} \) |
| 5 | \( 1 + 0.511T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 - 0.712T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 + 7.90T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 4.21T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.191590485303011686315180270734, −7.74535505597900602848372950616, −7.20221919795181399120600787121, −5.95058167067505990559357201581, −5.46741155524323261508723774278, −4.43243549843089303117808829156, −3.34004230992773473096512120147, −2.46047110588691229266107201439, −1.55175272164986652997979640006, −0.75261684477578761882321621814,
0.75261684477578761882321621814, 1.55175272164986652997979640006, 2.46047110588691229266107201439, 3.34004230992773473096512120147, 4.43243549843089303117808829156, 5.46741155524323261508723774278, 5.95058167067505990559357201581, 7.20221919795181399120600787121, 7.74535505597900602848372950616, 8.191590485303011686315180270734
