| L(s) = 1 | + 9.76·5-s + 136.·7-s − 653.·11-s + 250.·13-s − 249.·17-s − 1.75e3·19-s + 1.65e3·23-s − 3.02e3·25-s − 4.24e3·29-s + 8.98e3·31-s + 1.33e3·35-s − 6.00e3·37-s + 1.07e4·41-s + 1.00e4·43-s − 2.34e4·47-s + 1.87e3·49-s − 9.41e3·53-s − 6.38e3·55-s − 4.41e4·59-s − 2.24e4·61-s + 2.44e3·65-s − 3.60e4·67-s − 7.85e4·71-s + 6.13e4·73-s − 8.92e4·77-s + 2.74e4·79-s + 6.48e4·83-s + ⋯ |
| L(s) = 1 | + 0.174·5-s + 1.05·7-s − 1.62·11-s + 0.411·13-s − 0.209·17-s − 1.11·19-s + 0.652·23-s − 0.969·25-s − 0.937·29-s + 1.67·31-s + 0.184·35-s − 0.720·37-s + 0.998·41-s + 0.828·43-s − 1.55·47-s + 0.111·49-s − 0.460·53-s − 0.284·55-s − 1.65·59-s − 0.770·61-s + 0.0718·65-s − 0.979·67-s − 1.84·71-s + 1.34·73-s − 1.71·77-s + 0.495·79-s + 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9.76T + 3.12e3T^{2} \) |
| 7 | \( 1 - 136.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 250.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 249.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.75e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.41e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.61e4T + 8.58e9T^{2} \) |
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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
−10.60094467087870855400251550778, −9.356939044536823394773929127228, −8.207377920015571297224542618083, −7.72334893518644689185209611497, −6.30624198080997899257513307151, −5.24653281741732777704446569391, −4.36518321836775606719347977816, −2.76604282395064791780248206941, −1.64368616646693991397073991578, 0,
1.64368616646693991397073991578, 2.76604282395064791780248206941, 4.36518321836775606719347977816, 5.24653281741732777704446569391, 6.30624198080997899257513307151, 7.72334893518644689185209611497, 8.207377920015571297224542618083, 9.356939044536823394773929127228, 10.60094467087870855400251550778
