| L(s) = 1 | − 3·3-s − 21.6·7-s + 9·9-s − 21.8·11-s + 70.6·13-s − 54.6·17-s − 76.0·19-s + 64.8·21-s − 89.0·23-s − 27·27-s − 51.8·29-s − 110.·31-s + 65.4·33-s + 9.56·37-s − 211.·39-s + 317.·41-s + 374.·43-s − 347.·47-s + 124.·49-s + 163.·51-s + 159.·53-s + 228.·57-s − 693.·59-s − 684.·61-s − 194.·63-s + 559.·67-s + 267.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.16·7-s + 0.333·9-s − 0.597·11-s + 1.50·13-s − 0.779·17-s − 0.918·19-s + 0.673·21-s − 0.807·23-s − 0.192·27-s − 0.331·29-s − 0.637·31-s + 0.345·33-s + 0.0425·37-s − 0.870·39-s + 1.20·41-s + 1.32·43-s − 1.07·47-s + 0.361·49-s + 0.450·51-s + 0.413·53-s + 0.530·57-s − 1.53·59-s − 1.43·61-s − 0.388·63-s + 1.02·67-s + 0.466·69-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
| \(L(2)\) |
\(\approx\) |
\(0.7472557951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7472557951\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 21.6T + 343T^{2} \) |
| 11 | \( 1 + 21.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 89.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 51.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 9.56T + 5.06e4T^{2} \) |
| 41 | \( 1 - 317.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 347.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 684.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 559.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 517.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 745.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 341.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 706.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 171.T + 9.12e5T^{2} \) |
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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
−8.712210772571952101268509244509, −7.80743913880831401852078536025, −6.91228381067206980155710842794, −6.04129987835579334451323205481, −5.90864894826343269098826218303, −4.53022494924384189035255843555, −3.83612840607792923268670400538, −2.86575331210029018853654891066, −1.71383028610398382982922198281, −0.38669029637996410470696547195,
0.38669029637996410470696547195, 1.71383028610398382982922198281, 2.86575331210029018853654891066, 3.83612840607792923268670400538, 4.53022494924384189035255843555, 5.90864894826343269098826218303, 6.04129987835579334451323205481, 6.91228381067206980155710842794, 7.80743913880831401852078536025, 8.712210772571952101268509244509
