| L(s) = 1 | + 2.16·2-s + 3-s + 2.66·4-s − 2.38·5-s + 2.16·6-s + 2.35·7-s + 1.43·8-s + 9-s − 5.15·10-s − 11-s + 2.66·12-s + 5.08·14-s − 2.38·15-s − 2.22·16-s + 5.92·17-s + 2.16·18-s + 2.19·19-s − 6.36·20-s + 2.35·21-s − 2.16·22-s − 5.59·23-s + 1.43·24-s + 0.702·25-s + 27-s + 6.28·28-s + 6.94·29-s − 5.15·30-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 0.577·3-s + 1.33·4-s − 1.06·5-s + 0.881·6-s + 0.890·7-s + 0.508·8-s + 0.333·9-s − 1.63·10-s − 0.301·11-s + 0.769·12-s + 1.36·14-s − 0.616·15-s − 0.556·16-s + 1.43·17-s + 0.509·18-s + 0.503·19-s − 1.42·20-s + 0.514·21-s − 0.460·22-s − 1.16·23-s + 0.293·24-s + 0.140·25-s + 0.192·27-s + 1.18·28-s + 1.28·29-s − 0.941·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.553984397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.553984397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 5.59T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 - 9.44T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 - 9.50T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 - 9.85T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 - 8.61T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 0.229T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−7.985258067100389715805269728394, −7.53128352337206367805131405155, −6.61513779120917553654444340095, −5.75078458979489240802625951674, −5.07129024151568487283859144453, −4.31269187900321315825528901761, −3.89314090912049655313515071338, −3.03652171164816483396283492309, −2.36901779883640632389233588142, −1.01197542125506597841723445260,
1.01197542125506597841723445260, 2.36901779883640632389233588142, 3.03652171164816483396283492309, 3.89314090912049655313515071338, 4.31269187900321315825528901761, 5.07129024151568487283859144453, 5.75078458979489240802625951674, 6.61513779120917553654444340095, 7.53128352337206367805131405155, 7.985258067100389715805269728394
