| L(s) = 1 | − 7.76·2-s − 9·3-s + 28.2·4-s − 63.7·5-s + 69.8·6-s − 222.·7-s + 28.9·8-s + 81·9-s + 494.·10-s − 593.·11-s − 254.·12-s + 433.·13-s + 1.72e3·14-s + 573.·15-s − 1.12e3·16-s + 96.4·17-s − 628.·18-s + 1.54e3·19-s − 1.80e3·20-s + 2.00e3·21-s + 4.60e3·22-s − 955.·23-s − 260.·24-s + 939.·25-s − 3.36e3·26-s − 729·27-s − 6.28e3·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.883·4-s − 1.14·5-s + 0.792·6-s − 1.71·7-s + 0.159·8-s + 0.333·9-s + 1.56·10-s − 1.47·11-s − 0.510·12-s + 0.711·13-s + 2.35·14-s + 0.658·15-s − 1.10·16-s + 0.0809·17-s − 0.457·18-s + 0.980·19-s − 1.00·20-s + 0.990·21-s + 2.02·22-s − 0.376·23-s − 0.0923·24-s + 0.300·25-s − 0.976·26-s − 0.192·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 + 7.76T + 32T^{2} \) |
| 5 | \( 1 + 63.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 222.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 593.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 433.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 96.4T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 955.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.52e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.04e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.77e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.29e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.18e4T + 8.58e9T^{2} \) |
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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
−10.25238734680312483852415049777, −9.589365296526106832027417717890, −8.453442347065574965236416388989, −7.61981516324108332949402821189, −6.84791502251752480650435766717, −5.62526270893730902791688498134, −4.06436422788210240181461801186, −2.81917796270107166362557336256, −0.77237893414287613477710902548, 0,
0.77237893414287613477710902548, 2.81917796270107166362557336256, 4.06436422788210240181461801186, 5.62526270893730902791688498134, 6.84791502251752480650435766717, 7.61981516324108332949402821189, 8.453442347065574965236416388989, 9.589365296526106832027417717890, 10.25238734680312483852415049777
