| L(s) = 1 | − 3·3-s + 14.7·5-s − 31.4·7-s + 9·9-s − 11·11-s − 22.7·13-s − 44.1·15-s − 46.1·17-s + 4.11·19-s + 94.2·21-s + 163.·23-s + 91.2·25-s − 27·27-s + 188.·29-s + 210.·31-s + 33·33-s − 461.·35-s + 76.5·37-s + 68.1·39-s + 396.·41-s − 501.·43-s + 132.·45-s − 54.7·47-s + 643.·49-s + 138.·51-s − 338.·53-s − 161.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.31·5-s − 1.69·7-s + 0.333·9-s − 0.301·11-s − 0.484·13-s − 0.759·15-s − 0.657·17-s + 0.0496·19-s + 0.979·21-s + 1.48·23-s + 0.729·25-s − 0.192·27-s + 1.20·29-s + 1.22·31-s + 0.174·33-s − 2.23·35-s + 0.340·37-s + 0.279·39-s + 1.51·41-s − 1.78·43-s + 0.438·45-s − 0.170·47-s + 1.87·49-s + 0.379·51-s − 0.877·53-s − 0.396·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 - 14.7T + 125T^{2} \) |
| 7 | \( 1 + 31.4T + 343T^{2} \) |
| 13 | \( 1 + 22.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.11T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 338.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 565.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 820.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 299.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 175.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 377.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 841.T + 9.12e5T^{2} \) |
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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
−8.579543922565863800575491334426, −7.25474025076188390527265916867, −6.51720387714690591690413716425, −6.18984533869279141511714632799, −5.27846093357250755201574045954, −4.47796898347901354970849647735, −3.07332731810258926105659995186, −2.51144709292696402862363764563, −1.13718427157199356669876605765, 0,
1.13718427157199356669876605765, 2.51144709292696402862363764563, 3.07332731810258926105659995186, 4.47796898347901354970849647735, 5.27846093357250755201574045954, 6.18984533869279141511714632799, 6.51720387714690591690413716425, 7.25474025076188390527265916867, 8.579543922565863800575491334426
