| L(s) = 1 | + 24.4·2-s + 159.·3-s + 86.7·4-s − 2.02e3·5-s + 3.90e3·6-s − 1.04e4·8-s + 5.74e3·9-s − 4.96e4·10-s + 8.57e3·11-s + 1.38e4·12-s − 3.84e4·13-s − 3.23e5·15-s − 2.99e5·16-s − 3.01e5·17-s + 1.40e5·18-s − 9.16e5·19-s − 1.75e5·20-s + 2.09e5·22-s + 1.93e6·23-s − 1.65e6·24-s + 2.16e6·25-s − 9.40e5·26-s − 2.22e6·27-s + 4.52e6·29-s − 7.91e6·30-s + 2.94e5·31-s − 1.98e6·32-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.13·3-s + 0.169·4-s − 1.45·5-s + 1.22·6-s − 0.898·8-s + 0.292·9-s − 1.56·10-s + 0.176·11-s + 0.192·12-s − 0.373·13-s − 1.64·15-s − 1.14·16-s − 0.876·17-s + 0.315·18-s − 1.61·19-s − 0.245·20-s + 0.191·22-s + 1.44·23-s − 1.02·24-s + 1.10·25-s − 0.403·26-s − 0.804·27-s + 1.18·29-s − 1.78·30-s + 0.0572·31-s − 0.335·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 24.4T + 512T^{2} \) |
| 3 | \( 1 - 159.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.02e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 8.57e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.84e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.01e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.93e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.94e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.61e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.28e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.23e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.34e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.68e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.52e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.64e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.91e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.01e8T + 7.60e17T^{2} \) |
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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
−13.16927548650210332440335993989, −12.25903158647858739995000352270, −11.06289064902481601671290171574, −9.020936469942880128518483901756, −8.249629716026388802159711154733, −6.76286542906088885304889480848, −4.69934440364725434874394530060, −3.76380499114535424173374942626, −2.66892152906881678304081739186, 0,
2.66892152906881678304081739186, 3.76380499114535424173374942626, 4.69934440364725434874394530060, 6.76286542906088885304889480848, 8.249629716026388802159711154733, 9.020936469942880128518483901756, 11.06289064902481601671290171574, 12.25903158647858739995000352270, 13.16927548650210332440335993989
