| L(s) = 1 | − 2.25·3-s − 4.20·5-s + 3.20·7-s + 2.09·9-s − 11-s + 0.723·13-s + 9.48·15-s + 0.655·17-s + 19-s − 7.23·21-s − 2.68·23-s + 12.6·25-s + 2.05·27-s − 2.64·29-s + 9.48·31-s + 2.25·33-s − 13.4·35-s + 4.20·37-s − 1.63·39-s + 5.04·41-s − 10.7·43-s − 8.79·45-s − 4.84·47-s + 3.27·49-s − 1.47·51-s + 2.80·53-s + 4.20·55-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 1.88·5-s + 1.21·7-s + 0.696·9-s − 0.301·11-s + 0.200·13-s + 2.45·15-s + 0.159·17-s + 0.229·19-s − 1.57·21-s − 0.559·23-s + 2.53·25-s + 0.395·27-s − 0.490·29-s + 1.70·31-s + 0.392·33-s − 2.27·35-s + 0.692·37-s − 0.261·39-s + 0.787·41-s − 1.63·43-s − 1.31·45-s − 0.706·47-s + 0.468·49-s − 0.207·51-s + 0.384·53-s + 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 13 | \( 1 - 0.723T + 13T^{2} \) |
| 17 | \( 1 - 0.655T + 17T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 - 2.80T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 + 1.57T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.35T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 5.57T + 97T^{2} \) |
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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
−8.613225997701354845594977357892, −8.067805440033497911762511265690, −7.47617666144113073664759838545, −6.57354161027225128799642162289, −5.57242266851578808688940523140, −4.68220121489495181312543939159, −4.28648648207016159216283911580, −3.04037501706394497394622027914, −1.21132943358042517015228503552, 0,
1.21132943358042517015228503552, 3.04037501706394497394622027914, 4.28648648207016159216283911580, 4.68220121489495181312543939159, 5.57242266851578808688940523140, 6.57354161027225128799642162289, 7.47617666144113073664759838545, 8.067805440033497911762511265690, 8.613225997701354845594977357892
