| L(s) = 1 | + 0.930·2-s + 23.4·3-s − 127.·4-s − 494.·5-s + 21.8·6-s + 343·7-s − 237.·8-s − 1.63e3·9-s − 460.·10-s − 2.42e3·11-s − 2.98e3·12-s + 2.19e3·13-s + 319.·14-s − 1.15e4·15-s + 1.60e4·16-s + 3.89e3·17-s − 1.52e3·18-s − 4.51e3·19-s + 6.28e4·20-s + 8.04e3·21-s − 2.25e3·22-s + 6.60e4·23-s − 5.56e3·24-s + 1.66e5·25-s + 2.04e3·26-s − 8.96e4·27-s − 4.36e4·28-s + ⋯ |
| L(s) = 1 | + 0.0822·2-s + 0.501·3-s − 0.993·4-s − 1.76·5-s + 0.0412·6-s + 0.377·7-s − 0.163·8-s − 0.748·9-s − 0.145·10-s − 0.549·11-s − 0.497·12-s + 0.277·13-s + 0.0310·14-s − 0.887·15-s + 0.979·16-s + 0.192·17-s − 0.0615·18-s − 0.151·19-s + 1.75·20-s + 0.189·21-s − 0.0451·22-s + 1.13·23-s − 0.0822·24-s + 2.13·25-s + 0.0228·26-s − 0.876·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9164833254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9164833254\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 0.930T + 128T^{2} \) |
| 3 | \( 1 - 23.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 494.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.42e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.89e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.51e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.15e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.76e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.16e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.07e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.91e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 5.58e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.18e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.94e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.35e7T + 8.07e13T^{2} \) |
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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
−12.67566278465273841686950802739, −11.68272479102877733741487033121, −10.64726690301683206820412054030, −8.916971766152688153155814180860, −8.331863080677190488459615756733, −7.40421743811334569922958995200, −5.29135701171280410491793095896, −4.10714538922132723131700884267, −3.10108183696087487386120079086, −0.57955221890404332003864515888,
0.57955221890404332003864515888, 3.10108183696087487386120079086, 4.10714538922132723131700884267, 5.29135701171280410491793095896, 7.40421743811334569922958995200, 8.331863080677190488459615756733, 8.916971766152688153155814180860, 10.64726690301683206820412054030, 11.68272479102877733741487033121, 12.67566278465273841686950802739
