| L(s) = 1 | + 6.53·3-s + 21.4·5-s + 30.2·7-s + 15.7·9-s − 33.6·13-s + 140.·15-s − 4.20·17-s − 50.2·19-s + 197.·21-s − 27.2·23-s + 335.·25-s − 73.5·27-s + 24.3·29-s + 163.·31-s + 649.·35-s + 310.·37-s − 220.·39-s − 401.·41-s − 216.·43-s + 338.·45-s + 101.·47-s + 572.·49-s − 27.4·51-s + 235.·53-s − 328.·57-s − 126.·59-s − 255.·61-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 1.91·5-s + 1.63·7-s + 0.583·9-s − 0.718·13-s + 2.41·15-s − 0.0599·17-s − 0.606·19-s + 2.05·21-s − 0.246·23-s + 2.68·25-s − 0.523·27-s + 0.155·29-s + 0.946·31-s + 3.13·35-s + 1.38·37-s − 0.903·39-s − 1.52·41-s − 0.768·43-s + 1.11·45-s + 0.316·47-s + 1.66·49-s − 0.0754·51-s + 0.610·53-s − 0.762·57-s − 0.279·59-s − 0.535·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.613887007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.613887007\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 6.53T + 27T^{2} \) |
| 5 | \( 1 - 21.4T + 125T^{2} \) |
| 7 | \( 1 - 30.2T + 343T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.20T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 163.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 401.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 216.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 126.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 255.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 332.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 337.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 539.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 47.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 24.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 895.T + 9.12e5T^{2} \) |
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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
−9.624374107303278928567705914545, −8.691537389210294789733903424696, −8.269996425210103542352557107224, −7.24687549935114908601827560935, −6.14691179444438550931409326463, −5.21048226667606851775095020965, −4.43626282720056821931583804912, −2.81241039129955903295528222247, −2.13363924673164594045404210352, −1.43011373574650399672617812202,
1.43011373574650399672617812202, 2.13363924673164594045404210352, 2.81241039129955903295528222247, 4.43626282720056821931583804912, 5.21048226667606851775095020965, 6.14691179444438550931409326463, 7.24687549935114908601827560935, 8.269996425210103542352557107224, 8.691537389210294789733903424696, 9.624374107303278928567705914545
