| L(s) = 1 | + 3·3-s + 24.8·7-s + 9·9-s + 25.7·11-s − 60.6·13-s + 28.6·17-s − 86.6·19-s + 74.6·21-s − 52.3·23-s + 27·27-s + 6·29-s − 84.8·31-s + 77.2·33-s + 448.·37-s − 181.·39-s + 183.·41-s + 252·43-s + 41.9·47-s + 275.·49-s + 85.8·51-s + 228.·53-s − 259.·57-s − 179.·59-s + 480.·61-s + 223.·63-s + 855.·67-s − 157.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·7-s + 0.333·9-s + 0.705·11-s − 1.29·13-s + 0.408·17-s − 1.04·19-s + 0.775·21-s − 0.474·23-s + 0.192·27-s + 0.0384·29-s − 0.491·31-s + 0.407·33-s + 1.99·37-s − 0.746·39-s + 0.697·41-s + 0.893·43-s + 0.130·47-s + 0.802·49-s + 0.235·51-s + 0.592·53-s − 0.603·57-s − 0.396·59-s + 1.00·61-s + 0.447·63-s + 1.55·67-s − 0.273·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.458789369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.458789369\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 24.8T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 84.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 - 41.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 855.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 675.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 300.T + 9.12e5T^{2} \) |
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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
−8.509704024591743207397024058594, −7.86772947776091982134007289724, −7.32962117792787068376151573614, −6.34090657007773506850608972132, −5.35456762986367189730602596077, −4.49232629562224694080658463673, −3.94402541202470043841885110282, −2.56954183383981135724820815684, −1.94303175172602254029880458212, −0.816028932753987189817538213229,
0.816028932753987189817538213229, 1.94303175172602254029880458212, 2.56954183383981135724820815684, 3.94402541202470043841885110282, 4.49232629562224694080658463673, 5.35456762986367189730602596077, 6.34090657007773506850608972132, 7.32962117792787068376151573614, 7.86772947776091982134007289724, 8.509704024591743207397024058594
