L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 455.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s + 3.64e3·10-s − 5.88e3·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 1.23e4·15-s + 4.09e3·16-s + 3.93e3·17-s − 5.83e3·18-s + 4.33e4·19-s − 2.91e4·20-s − 9.26e3·21-s + 4.70e4·22-s − 2.54e4·23-s − 1.38e4·24-s + 1.29e5·25-s + 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.63·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.15·10-s − 1.33·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.941·15-s + 0.250·16-s + 0.194·17-s − 0.235·18-s + 1.44·19-s − 0.815·20-s − 0.218·21-s + 0.942·22-s − 0.436·23-s − 0.204·24-s + 1.66·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
good | 5 | \( 1 + 455.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.88e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.93e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.33e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.54e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.59e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.88e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.35e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.17e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.63e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.57e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.23e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.89e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.55e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.58e7T + 8.07e13T^{2} \) |
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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
−9.160755657880619390401598016720, −8.059288578178789894704723737193, −7.75075456172237101997472618912, −7.03583294208605317915766778215, −5.55320869126221193085207050732, −4.31519418879324309142069293965, −3.28484110108141654983384238202, −2.54892257798947194052000422937, −0.909943641939862083746555517512, 0,
0.909943641939862083746555517512, 2.54892257798947194052000422937, 3.28484110108141654983384238202, 4.31519418879324309142069293965, 5.55320869126221193085207050732, 7.03583294208605317915766778215, 7.75075456172237101997472618912, 8.059288578178789894704723737193, 9.160755657880619390401598016720