| L(s) = 1 | − 29.5·3-s − 41.2·5-s − 77.0·7-s + 632.·9-s − 121·11-s + 637.·13-s + 1.22e3·15-s + 1.42e3·17-s − 1.71e3·19-s + 2.28e3·21-s + 4.19e3·23-s − 1.42e3·25-s − 1.15e4·27-s − 878.·29-s − 6.76e3·31-s + 3.58e3·33-s + 3.17e3·35-s + 4.00e3·37-s − 1.88e4·39-s − 1.36e4·41-s + 1.62e3·43-s − 2.60e4·45-s + 1.71e4·47-s − 1.08e4·49-s − 4.21e4·51-s − 1.51e4·53-s + 4.98e3·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s − 0.737·5-s − 0.594·7-s + 2.60·9-s − 0.301·11-s + 1.04·13-s + 1.40·15-s + 1.19·17-s − 1.09·19-s + 1.12·21-s + 1.65·23-s − 0.455·25-s − 3.04·27-s − 0.193·29-s − 1.26·31-s + 0.572·33-s + 0.438·35-s + 0.481·37-s − 1.98·39-s − 1.27·41-s + 0.134·43-s − 1.91·45-s + 1.13·47-s − 0.646·49-s − 2.26·51-s − 0.740·53-s + 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5292689799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5292689799\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 29.5T + 243T^{2} \) |
| 5 | \( 1 + 41.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 77.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 637.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 878.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.36e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.56e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.44e4T + 8.58e9T^{2} \) |
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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
−10.04420913443212423301414820044, −8.885869778220952630094291083841, −7.63655772833727064654680325082, −6.88595146591814492675961883493, −6.03161785069788146247682990004, −5.35898307195928696563707900797, −4.31242914270213760778531967310, −3.40926946788064908173792372797, −1.42406529198683027605154665108, −0.40807781282416092667454215704,
0.40807781282416092667454215704, 1.42406529198683027605154665108, 3.40926946788064908173792372797, 4.31242914270213760778531967310, 5.35898307195928696563707900797, 6.03161785069788146247682990004, 6.88595146591814492675961883493, 7.63655772833727064654680325082, 8.885869778220952630094291083841, 10.04420913443212423301414820044
