| L(s) = 1 | − 9·3-s + 36.1·5-s + 186.·7-s + 81·9-s − 369.·11-s − 306.·13-s − 325.·15-s − 374.·17-s − 361·19-s − 1.68e3·21-s − 826.·23-s − 1.81e3·25-s − 729·27-s + 1.17e3·29-s − 5.81e3·31-s + 3.32e3·33-s + 6.75e3·35-s − 4.38e3·37-s + 2.75e3·39-s − 3.96e3·41-s + 2.32e3·43-s + 2.92e3·45-s + 1.86e4·47-s + 1.80e4·49-s + 3.36e3·51-s + 6.81e3·53-s − 1.33e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.646·5-s + 1.44·7-s + 0.333·9-s − 0.920·11-s − 0.502·13-s − 0.373·15-s − 0.314·17-s − 0.229·19-s − 0.831·21-s − 0.325·23-s − 0.581·25-s − 0.192·27-s + 0.259·29-s − 1.08·31-s + 0.531·33-s + 0.931·35-s − 0.526·37-s + 0.290·39-s − 0.368·41-s + 0.191·43-s + 0.215·45-s + 1.22·47-s + 1.07·49-s + 0.181·51-s + 0.333·53-s − 0.595·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 - 36.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 186.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 369.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 306.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 374.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 826.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.62e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.40e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.38e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e5T + 8.58e9T^{2} \) |
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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
−10.05667536021959608745482755191, −8.891713561105634244375455500159, −7.923297915137644043761536557309, −7.10698920223541555190643605880, −5.76952756028760858020708980244, −5.18326853038587217805381092783, −4.21934320185595415776978269793, −2.41924350077698235206189116869, −1.52103326359286874463249518412, 0,
1.52103326359286874463249518412, 2.41924350077698235206189116869, 4.21934320185595415776978269793, 5.18326853038587217805381092783, 5.76952756028760858020708980244, 7.10698920223541555190643605880, 7.923297915137644043761536557309, 8.891713561105634244375455500159, 10.05667536021959608745482755191
