| L(s) = 1 | − 2.60·2-s − 1.18·3-s + 4.77·4-s + 3.07·6-s + 3.53·7-s − 7.21·8-s − 1.60·9-s + 2.94·11-s − 5.64·12-s + 4.01·13-s − 9.20·14-s + 9.23·16-s − 17-s + 4.17·18-s − 6.97·19-s − 4.18·21-s − 7.66·22-s − 6.12·23-s + 8.53·24-s − 10.4·26-s + 5.44·27-s + 16.8·28-s + 5.30·29-s + 6.49·31-s − 9.59·32-s − 3.48·33-s + 2.60·34-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.682·3-s + 2.38·4-s + 1.25·6-s + 1.33·7-s − 2.55·8-s − 0.534·9-s + 0.888·11-s − 1.62·12-s + 1.11·13-s − 2.45·14-s + 2.30·16-s − 0.242·17-s + 0.982·18-s − 1.60·19-s − 0.912·21-s − 1.63·22-s − 1.27·23-s + 1.74·24-s − 2.04·26-s + 1.04·27-s + 3.18·28-s + 0.984·29-s + 1.16·31-s − 1.69·32-s − 0.606·33-s + 0.446·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5473207215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5473207215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 - 2.00T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−10.94805070342518662444957674120, −10.48113946114586763958242639779, −9.179829265380621468720940348191, −8.422462251088697546291467715765, −7.962402113983155982404201152036, −6.52136385484807824788455224299, −6.02983365902527318815063093902, −4.33516982401709700523998025002, −2.25703778911600787632433421835, −0.989009460232569672358177642769,
0.989009460232569672358177642769, 2.25703778911600787632433421835, 4.33516982401709700523998025002, 6.02983365902527318815063093902, 6.52136385484807824788455224299, 7.962402113983155982404201152036, 8.422462251088697546291467715765, 9.179829265380621468720940348191, 10.48113946114586763958242639779, 10.94805070342518662444957674120
