| L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 0.828·5-s − 2.41·6-s + 0.635·7-s + 8-s + 2.83·9-s + 0.828·10-s + 4.76·11-s − 2.41·12-s + 1.33·13-s + 0.635·14-s − 2.00·15-s + 16-s + 4.13·17-s + 2.83·18-s − 19-s + 0.828·20-s − 1.53·21-s + 4.76·22-s + 8.13·23-s − 2.41·24-s − 4.31·25-s + 1.33·26-s + 0.398·27-s + 0.635·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.370·5-s − 0.986·6-s + 0.240·7-s + 0.353·8-s + 0.945·9-s + 0.261·10-s + 1.43·11-s − 0.697·12-s + 0.371·13-s + 0.169·14-s − 0.516·15-s + 0.250·16-s + 1.00·17-s + 0.668·18-s − 0.229·19-s + 0.185·20-s − 0.334·21-s + 1.01·22-s + 1.69·23-s − 0.493·24-s − 0.862·25-s + 0.262·26-s + 0.0766·27-s + 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.829704440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829704440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.828T + 5T^{2} \) |
| 7 | \( 1 - 0.635T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 37 | \( 1 - 0.803T + 37T^{2} \) |
| 41 | \( 1 - 9.53T + 41T^{2} \) |
| 43 | \( 1 - 0.674T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 6.64T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.475T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 - 1.33T + 97T^{2} \) |
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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
−7.51703210786051623641674521202, −6.82692211273569402095511275736, −6.24953095426284748907806136315, −5.78033255879358170730347539569, −5.15357666156920536334287579599, −4.43028708975778080937283263424, −3.75687282833085226142963662833, −2.78659082462578558498489221146, −1.51678126261319846572492966217, −0.901257334840586146491266477496,
0.901257334840586146491266477496, 1.51678126261319846572492966217, 2.78659082462578558498489221146, 3.75687282833085226142963662833, 4.43028708975778080937283263424, 5.15357666156920536334287579599, 5.78033255879358170730347539569, 6.24953095426284748907806136315, 6.82692211273569402095511275736, 7.51703210786051623641674521202
