| L(s) = 1 | − 3.60·2-s + 4.99·4-s − 9·7-s + 10.8·8-s − 14.4·11-s + 34·13-s + 32.4·14-s − 79·16-s + 7.21·17-s − 101·19-s + 51.9·22-s + 108.·23-s − 122.·26-s − 44.9·28-s + 165.·29-s − 3·31-s + 198.·32-s − 25.9·34-s − 67·37-s + 364.·38-s − 201.·41-s − 137·43-s − 72.1·44-s − 390·46-s + 115.·47-s − 262·49-s + 169.·52-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.624·4-s − 0.485·7-s + 0.478·8-s − 0.395·11-s + 0.725·13-s + 0.619·14-s − 1.23·16-s + 0.102·17-s − 1.21·19-s + 0.503·22-s + 0.980·23-s − 0.924·26-s − 0.303·28-s + 1.06·29-s − 0.0173·31-s + 1.09·32-s − 0.131·34-s − 0.297·37-s + 1.55·38-s − 0.769·41-s − 0.485·43-s − 0.247·44-s − 1.25·46-s + 0.358·47-s − 0.763·49-s + 0.453·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.60T + 8T^{2} \) |
| 7 | \( 1 + 9T + 343T^{2} \) |
| 11 | \( 1 + 14.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.21T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67T + 5.06e4T^{2} \) |
| 41 | \( 1 + 201.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 137T + 7.95e4T^{2} \) |
| 47 | \( 1 - 115.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 677.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 872.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 563T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 786.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 503T + 3.89e5T^{2} \) |
| 79 | \( 1 - 615T + 4.93e5T^{2} \) |
| 83 | \( 1 + 237.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 973.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 19T + 9.12e5T^{2} \) |
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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
−9.653615177128799339858546407925, −8.658644378013392503406004328059, −8.327029873538318416435291916222, −7.12933377947154105195811115838, −6.45137704185909663750377940568, −5.14299765028413885321342381311, −3.95009122690299283750359061872, −2.56876196431752541391331981678, −1.21980592044597267614797041905, 0,
1.21980592044597267614797041905, 2.56876196431752541391331981678, 3.95009122690299283750359061872, 5.14299765028413885321342381311, 6.45137704185909663750377940568, 7.12933377947154105195811115838, 8.327029873538318416435291916222, 8.658644378013392503406004328059, 9.653615177128799339858546407925
