| L(s) = 1 | − 8·2-s − 44·3-s + 64·4-s − 125·5-s + 352·6-s + 524·7-s − 512·8-s + 1.20e3·9-s + 1.00e3·10-s − 2.81e3·12-s − 4.19e3·14-s + 5.50e3·15-s + 4.09e3·16-s − 9.65e3·18-s − 8.00e3·20-s − 2.30e4·21-s + 1.53e4·23-s + 2.25e4·24-s + 1.56e4·25-s − 2.10e4·27-s + 3.35e4·28-s + 4.48e4·29-s − 4.40e4·30-s − 3.27e4·32-s − 6.55e4·35-s + 7.72e4·36-s + 6.40e4·40-s + ⋯ |
| L(s) = 1 | − 2-s − 1.62·3-s + 4-s − 5-s + 1.62·6-s + 1.52·7-s − 8-s + 1.65·9-s + 10-s − 1.62·12-s − 1.52·14-s + 1.62·15-s + 16-s − 1.65·18-s − 20-s − 2.48·21-s + 1.26·23-s + 1.62·24-s + 25-s − 1.06·27-s + 1.52·28-s + 1.83·29-s − 1.62·30-s − 32-s − 1.52·35-s + 1.65·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.5341878580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5341878580\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 5 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 44 T + p^{6} T^{2} \) |
| 7 | \( 1 - 524 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 - 15356 T + p^{6} T^{2} \) |
| 29 | \( 1 - 44858 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 + 74338 T + p^{6} T^{2} \) |
| 43 | \( 1 + 17404 T + p^{6} T^{2} \) |
| 47 | \( 1 - 26444 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 452342 T + p^{6} T^{2} \) |
| 67 | \( 1 + 1276 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( 1 - 1131716 T + p^{6} T^{2} \) |
| 89 | \( 1 - 511058 T + p^{6} T^{2} \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−17.22295989848122149650783756062, −16.13742806024158998255784262244, −14.98765456448803144049581338089, −12.15101653989214036314938812078, −11.40292932966400066928821279626, −10.59448415266923782408622638073, −8.314417512756925902264958020299, −6.91103528665437821657493746827, −4.97686906242514601602353639083, −0.915759703166410749817117910843,
0.915759703166410749817117910843, 4.97686906242514601602353639083, 6.91103528665437821657493746827, 8.314417512756925902264958020299, 10.59448415266923782408622638073, 11.40292932966400066928821279626, 12.15101653989214036314938812078, 14.98765456448803144049581338089, 16.13742806024158998255784262244, 17.22295989848122149650783756062
