| L(s) = 1 | + 3·3-s + 6.70·5-s − 7·7-s + 9·9-s − 0.704·11-s − 35.4·13-s + 20.1·15-s + 26.7·17-s − 30.8·19-s − 21·21-s − 5.88·23-s − 80.0·25-s + 27·27-s − 252.·29-s + 37.6·31-s − 2.11·33-s − 46.9·35-s + 139.·37-s − 106.·39-s − 46.7·41-s + 254.·43-s + 60.3·45-s − 607.·47-s + 49·49-s + 80.1·51-s − 298.·53-s − 4.72·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.599·5-s − 0.377·7-s + 0.333·9-s − 0.0193·11-s − 0.755·13-s + 0.346·15-s + 0.380·17-s − 0.372·19-s − 0.218·21-s − 0.0533·23-s − 0.640·25-s + 0.192·27-s − 1.61·29-s + 0.218·31-s − 0.0111·33-s − 0.226·35-s + 0.621·37-s − 0.436·39-s − 0.177·41-s + 0.901·43-s + 0.199·45-s − 1.88·47-s + 0.142·49-s + 0.219·51-s − 0.774·53-s − 0.0115·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 6.70T + 125T^{2} \) |
| 11 | \( 1 + 0.704T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 5.88T + 1.21e4T^{2} \) |
| 29 | \( 1 + 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 46.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 607.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 178.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 390.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 349.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 646.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 278.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 154.T + 9.12e5T^{2} \) |
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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
−8.995926815142668447082834777905, −7.972274782383951708226289312716, −7.33053036747255215976524340031, −6.33724415184341017828804961206, −5.55741749274650344830593587081, −4.51049618450207524276787663175, −3.48672196550023578231513332047, −2.51135983671963256793167652990, −1.59546171071849534203617451820, 0,
1.59546171071849534203617451820, 2.51135983671963256793167652990, 3.48672196550023578231513332047, 4.51049618450207524276787663175, 5.55741749274650344830593587081, 6.33724415184341017828804961206, 7.33053036747255215976524340031, 7.972274782383951708226289312716, 8.995926815142668447082834777905
