| L(s) = 1 | + 2.02·2-s + 2.91·3-s + 2.09·4-s + 5.89·6-s + 3.21·7-s + 0.185·8-s + 5.49·9-s + 6.09·12-s − 0.648·13-s + 6.49·14-s − 3.80·16-s + 1.18·17-s + 11.1·18-s + 1.89·19-s + 9.36·21-s − 4.35·23-s + 0.540·24-s − 1.31·26-s + 7.27·27-s + 6.71·28-s − 4.16·29-s + 7.89·31-s − 8.07·32-s + 2.40·34-s + 11.4·36-s + 2.05·37-s + 3.82·38-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.68·3-s + 1.04·4-s + 2.40·6-s + 1.21·7-s + 0.0656·8-s + 1.83·9-s + 1.76·12-s − 0.179·13-s + 1.73·14-s − 0.952·16-s + 0.288·17-s + 2.62·18-s + 0.433·19-s + 2.04·21-s − 0.909·23-s + 0.110·24-s − 0.257·26-s + 1.40·27-s + 1.26·28-s − 0.773·29-s + 1.41·31-s − 1.42·32-s + 0.412·34-s + 1.91·36-s + 0.338·37-s + 0.620·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.944253147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.944253147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 2.91T + 3T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 13 | \( 1 + 0.648T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 - 2.05T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 + 7.78T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.425209123601998437044266511676, −8.080623140455987146881088248771, −7.29316581834186342207489596935, −6.39843845939932196655272605325, −5.32501776200887088547746467934, −4.67321789489903687015581569605, −3.93689380624025124257151060088, −3.24175594449956133274113093564, −2.40406063764006427970445090653, −1.62240957006508621956999792310,
1.62240957006508621956999792310, 2.40406063764006427970445090653, 3.24175594449956133274113093564, 3.93689380624025124257151060088, 4.67321789489903687015581569605, 5.32501776200887088547746467934, 6.39843845939932196655272605325, 7.29316581834186342207489596935, 8.080623140455987146881088248771, 8.425209123601998437044266511676
