| L(s) = 1 | + 7.92·3-s + 14.8·5-s + 3.07·7-s + 35.8·9-s − 5.35·13-s + 117.·15-s + 41.2·17-s + 139.·19-s + 24.3·21-s + 111.·23-s + 95.7·25-s + 70.2·27-s + 24.9·29-s − 31.4·31-s + 45.6·35-s + 13.1·37-s − 42.4·39-s − 261.·41-s − 57.7·43-s + 532.·45-s + 343.·47-s − 333.·49-s + 326.·51-s − 342.·53-s + 1.10e3·57-s − 88.3·59-s − 738.·61-s + ⋯ |
| L(s) = 1 | + 1.52·3-s + 1.32·5-s + 0.165·7-s + 1.32·9-s − 0.114·13-s + 2.02·15-s + 0.588·17-s + 1.68·19-s + 0.253·21-s + 1.00·23-s + 0.765·25-s + 0.500·27-s + 0.160·29-s − 0.182·31-s + 0.220·35-s + 0.0583·37-s − 0.174·39-s − 0.994·41-s − 0.204·43-s + 1.76·45-s + 1.06·47-s − 0.972·49-s + 0.897·51-s − 0.888·53-s + 2.57·57-s − 0.194·59-s − 1.55·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.059344908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.059344908\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 7.92T + 27T^{2} \) |
| 5 | \( 1 - 14.8T + 125T^{2} \) |
| 7 | \( 1 - 3.07T + 343T^{2} \) |
| 13 | \( 1 + 5.35T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 88.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 207.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 441.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−9.073478652187552645013872380145, −8.069314004067824914806206309027, −7.48604242322138054949041497653, −6.57455903811688613081368899303, −5.54575749685884778098253193593, −4.84327047937048172837008824224, −3.46855031830109943323308114556, −2.91515309100638842056793562219, −1.96215967843596157881545104610, −1.17092688024732954744769123660,
1.17092688024732954744769123660, 1.96215967843596157881545104610, 2.91515309100638842056793562219, 3.46855031830109943323308114556, 4.84327047937048172837008824224, 5.54575749685884778098253193593, 6.57455903811688613081368899303, 7.48604242322138054949041497653, 8.069314004067824914806206309027, 9.073478652187552645013872380145
