| L(s) = 1 | − 6·2-s + 42·3-s − 92·4-s + 84·5-s − 252·6-s + 1.32e3·8-s − 423·9-s − 504·10-s − 5.56e3·11-s − 3.86e3·12-s + 5.15e3·13-s + 3.52e3·15-s + 3.85e3·16-s + 1.39e4·17-s + 2.53e3·18-s − 5.53e4·19-s − 7.72e3·20-s + 3.34e4·22-s − 9.12e4·23-s + 5.54e4·24-s − 7.10e4·25-s − 3.09e4·26-s − 1.09e5·27-s + 4.16e4·29-s − 2.11e4·30-s − 1.50e5·31-s − 1.92e5·32-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 0.898·3-s − 0.718·4-s + 0.300·5-s − 0.476·6-s + 0.911·8-s − 0.193·9-s − 0.159·10-s − 1.26·11-s − 0.645·12-s + 0.650·13-s + 0.269·15-s + 0.235·16-s + 0.690·17-s + 0.102·18-s − 1.85·19-s − 0.216·20-s + 0.668·22-s − 1.56·23-s + 0.818·24-s − 0.909·25-s − 0.344·26-s − 1.07·27-s + 0.316·29-s − 0.143·30-s − 0.906·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + p^{7} T^{2} \) |
| 3 | \( 1 - 14 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 84 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5568 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5152 T + p^{7} T^{2} \) |
| 17 | \( 1 - 13986 T + p^{7} T^{2} \) |
| 19 | \( 1 + 55370 T + p^{7} T^{2} \) |
| 23 | \( 1 + 91272 T + p^{7} T^{2} \) |
| 29 | \( 1 - 41610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 150332 T + p^{7} T^{2} \) |
| 37 | \( 1 + 136366 T + p^{7} T^{2} \) |
| 41 | \( 1 - 510258 T + p^{7} T^{2} \) |
| 43 | \( 1 + 172072 T + p^{7} T^{2} \) |
| 47 | \( 1 - 519036 T + p^{7} T^{2} \) |
| 53 | \( 1 + 59202 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1979250 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2988748 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2409404 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1504512 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1821022 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1669240 T + p^{7} T^{2} \) |
| 83 | \( 1 + 696738 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5558490 T + p^{7} T^{2} \) |
| 97 | \( 1 + 101822 p T + p^{7} T^{2} \) |
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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
−13.66249757393701650923927235805, −12.71920443633866476644313459251, −10.74466376290564794186139225827, −9.730998251008340509899866794607, −8.514199905656442900204160944455, −7.88781114289792529677732820903, −5.70376579669499892911437543617, −3.91868741583730322731664960762, −2.13237537609651696646917143737, 0,
2.13237537609651696646917143737, 3.91868741583730322731664960762, 5.70376579669499892911437543617, 7.88781114289792529677732820903, 8.514199905656442900204160944455, 9.730998251008340509899866794607, 10.74466376290564794186139225827, 12.71920443633866476644313459251, 13.66249757393701650923927235805
