| L(s) = 1 | + 2.64·2-s + 3-s + 5.01·4-s − 0.130·5-s + 2.64·6-s + 0.903·7-s + 7.99·8-s + 9-s − 0.346·10-s + 5.64·11-s + 5.01·12-s + 0.675·13-s + 2.39·14-s − 0.130·15-s + 11.1·16-s − 6.72·17-s + 2.64·18-s + 19-s − 0.657·20-s + 0.903·21-s + 14.9·22-s − 9.38·23-s + 7.99·24-s − 4.98·25-s + 1.78·26-s + 27-s + 4.53·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.50·4-s − 0.0585·5-s + 1.08·6-s + 0.341·7-s + 2.82·8-s + 0.333·9-s − 0.109·10-s + 1.70·11-s + 1.44·12-s + 0.187·13-s + 0.639·14-s − 0.0338·15-s + 2.78·16-s − 1.63·17-s + 0.624·18-s + 0.229·19-s − 0.146·20-s + 0.197·21-s + 3.18·22-s − 1.95·23-s + 1.63·24-s − 0.996·25-s + 0.350·26-s + 0.192·27-s + 0.856·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.218175428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.218175428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 + 0.130T + 5T^{2} \) |
| 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 0.675T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 23 | \( 1 + 9.38T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 - 0.635T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 + 1.45T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 2.20T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 9.87T + 97T^{2} \) |
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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
−8.529779034467451497389247523382, −7.81318053055571550506851714687, −6.74982907793504072021071794152, −6.45825682379263282957528290741, −5.61696677952311050521418436324, −4.43787811029399701882192698859, −4.16368091886786023345313853199, −3.42652716031056672713290114318, −2.28446194332281133839537209649, −1.63796620331703370564090160437,
1.63796620331703370564090160437, 2.28446194332281133839537209649, 3.42652716031056672713290114318, 4.16368091886786023345313853199, 4.43787811029399701882192698859, 5.61696677952311050521418436324, 6.45825682379263282957528290741, 6.74982907793504072021071794152, 7.81318053055571550506851714687, 8.529779034467451497389247523382
