L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 176.·5-s − 216·6-s − 148.·7-s − 512·8-s + 729·9-s − 1.41e3·10-s + 994.·11-s + 1.72e3·12-s − 1.09e4·13-s + 1.18e3·14-s + 4.76e3·15-s + 4.09e3·16-s + 9.14e3·17-s − 5.83e3·18-s + 3.32e4·19-s + 1.12e4·20-s − 4.00e3·21-s − 7.95e3·22-s − 2.28e4·23-s − 1.38e4·24-s − 4.70e4·25-s + 8.78e4·26-s + 1.96e4·27-s − 9.49e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.631·5-s − 0.408·6-s − 0.163·7-s − 0.353·8-s + 0.333·9-s − 0.446·10-s + 0.225·11-s + 0.288·12-s − 1.38·13-s + 0.115·14-s + 0.364·15-s + 0.250·16-s + 0.451·17-s − 0.235·18-s + 1.11·19-s + 0.315·20-s − 0.0943·21-s − 0.159·22-s − 0.390·23-s − 0.204·24-s − 0.601·25-s + 0.980·26-s + 0.192·27-s − 0.0817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 5 | \( 1 - 176.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 148.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 994.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.09e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 9.14e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.32e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.28e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.05e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.34e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.70e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.54e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.81e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.28e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 4.68e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.04e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.24e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.71e6T + 8.07e13T^{2} \) |
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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
−9.730456349868614047570747017826, −9.113595496355845029914584627069, −7.892446476793421500021777091944, −7.26183275104742251929163101896, −6.09515979355996916705850098847, −4.98049692652732538738593554217, −3.45059688085521716860597355868, −2.38951863956471379123836920100, −1.42977823233667579174145630406, 0,
1.42977823233667579174145630406, 2.38951863956471379123836920100, 3.45059688085521716860597355868, 4.98049692652732538738593554217, 6.09515979355996916705850098847, 7.26183275104742251929163101896, 7.892446476793421500021777091944, 9.113595496355845029914584627069, 9.730456349868614047570747017826