| L(s) = 1 | + 4.44·2-s + 0.537·3-s + 11.7·4-s + 17.2·5-s + 2.38·6-s − 6.40·7-s + 16.6·8-s − 26.7·9-s + 76.6·10-s + 55.3·11-s + 6.31·12-s + 58.6·13-s − 28.4·14-s + 9.27·15-s − 20.0·16-s − 118.·18-s − 91.1·19-s + 202.·20-s − 3.44·21-s + 245.·22-s + 120.·23-s + 8.94·24-s + 172.·25-s + 260.·26-s − 28.8·27-s − 75.2·28-s − 215.·29-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 0.103·3-s + 1.46·4-s + 1.54·5-s + 0.162·6-s − 0.345·7-s + 0.735·8-s − 0.989·9-s + 2.42·10-s + 1.51·11-s + 0.151·12-s + 1.25·13-s − 0.543·14-s + 0.159·15-s − 0.312·16-s − 1.55·18-s − 1.10·19-s + 2.26·20-s − 0.0357·21-s + 2.38·22-s + 1.09·23-s + 0.0760·24-s + 1.38·25-s + 1.96·26-s − 0.205·27-s − 0.507·28-s − 1.38·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.535244584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.535244584\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 4.44T + 8T^{2} \) |
| 3 | \( 1 - 0.537T + 27T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 7 | \( 1 + 6.40T + 343T^{2} \) |
| 11 | \( 1 - 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 91.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 17.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 8.40T + 5.06e4T^{2} \) |
| 41 | \( 1 - 99.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 81.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 195.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 536.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 514.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 704.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 184.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 34.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 647.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 256.T + 9.12e5T^{2} \) |
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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
−11.49338361138371671777379276061, −10.81854039615553149836971818937, −9.331517253300390286808212312947, −8.807182301861213003452024860516, −6.71445934552185223431436597325, −6.12687288853462850024465949343, −5.49285693440540549017670811635, −4.07074127655670534434787291870, −3.00654783427400569761983311192, −1.71846238001440569590011369865,
1.71846238001440569590011369865, 3.00654783427400569761983311192, 4.07074127655670534434787291870, 5.49285693440540549017670811635, 6.12687288853462850024465949343, 6.71445934552185223431436597325, 8.807182301861213003452024860516, 9.331517253300390286808212312947, 10.81854039615553149836971818937, 11.49338361138371671777379276061
