L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 294.·5-s − 216·6-s + 995.·7-s − 512·8-s + 729·9-s − 2.35e3·10-s + 1.86e3·11-s + 1.72e3·12-s + 1.31e4·13-s − 7.96e3·14-s + 7.94e3·15-s + 4.09e3·16-s + 7.52e3·17-s − 5.83e3·18-s + 4.33e3·19-s + 1.88e4·20-s + 2.68e4·21-s − 1.49e4·22-s + 1.21e4·23-s − 1.38e4·24-s + 8.51e3·25-s − 1.05e5·26-s + 1.96e4·27-s + 6.37e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.05·5-s − 0.408·6-s + 1.09·7-s − 0.353·8-s + 0.333·9-s − 0.744·10-s + 0.423·11-s + 0.288·12-s + 1.66·13-s − 0.775·14-s + 0.608·15-s + 0.250·16-s + 0.371·17-s − 0.235·18-s + 0.145·19-s + 0.526·20-s + 0.633·21-s − 0.299·22-s + 0.208·23-s − 0.204·24-s + 0.109·25-s − 1.17·26-s + 0.192·27-s + 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.930114744\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.930114744\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
good | 5 | \( 1 - 294.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 995.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.86e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.31e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.52e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.33e3T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.25e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.62e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.02e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.28e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.58e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.36e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.39e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.92e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.76e6T + 8.07e13T^{2} \) |
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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
−11.53457333073121410160270162741, −10.70998396931441179033286862792, −9.595370332401300398119147600748, −8.747214727129559285709225449606, −7.901000958868625983169315633107, −6.53559094367019622429566924085, −5.37508666585563581216716427079, −3.62143155394359994126541624420, −1.96418293327540197224196345164, −1.23990081113545917004334436514,
1.23990081113545917004334436514, 1.96418293327540197224196345164, 3.62143155394359994126541624420, 5.37508666585563581216716427079, 6.53559094367019622429566924085, 7.901000958868625983169315633107, 8.747214727129559285709225449606, 9.595370332401300398119147600748, 10.70998396931441179033286862792, 11.53457333073121410160270162741