| L(s) = 1 | + 1.83·2-s − 0.438·3-s + 1.38·4-s − 2.41·5-s − 0.806·6-s + 4.05·7-s − 1.13·8-s − 2.80·9-s − 4.43·10-s − 0.114·11-s − 0.606·12-s − 5.97·13-s + 7.46·14-s + 1.05·15-s − 4.85·16-s − 17-s − 5.16·18-s + 2.30·19-s − 3.33·20-s − 1.78·21-s − 0.211·22-s − 3.40·23-s + 0.498·24-s + 0.811·25-s − 10.9·26-s + 2.54·27-s + 5.60·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s − 0.253·3-s + 0.690·4-s − 1.07·5-s − 0.329·6-s + 1.53·7-s − 0.401·8-s − 0.935·9-s − 1.40·10-s − 0.0346·11-s − 0.175·12-s − 1.65·13-s + 1.99·14-s + 0.273·15-s − 1.21·16-s − 0.242·17-s − 1.21·18-s + 0.529·19-s − 0.744·20-s − 0.388·21-s − 0.0450·22-s − 0.709·23-s + 0.101·24-s + 0.162·25-s − 2.15·26-s + 0.490·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.438T + 3T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 + 0.114T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 + 0.455T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 - 0.738T + 97T^{2} \) |
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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
−9.027320951618355255684734441804, −8.188758044868957934452341483549, −7.57380795653186073012103801424, −6.61387541114375963170192405561, −5.33040776613010798792316212653, −5.03339924833740230733670530588, −4.23924718363276869748913898304, −3.24867482638556864766224283957, −2.15542654790962454149692938703, 0,
2.15542654790962454149692938703, 3.24867482638556864766224283957, 4.23924718363276869748913898304, 5.03339924833740230733670530588, 5.33040776613010798792316212653, 6.61387541114375963170192405561, 7.57380795653186073012103801424, 8.188758044868957934452341483549, 9.027320951618355255684734441804
