| L(s) = 1 | + 2.29·2-s + 1.48·3-s + 3.25·4-s + 3.39·6-s + 1.39·7-s + 2.88·8-s − 0.801·9-s + 1.87·11-s + 4.82·12-s − 2.05·13-s + 3.18·14-s + 0.0915·16-s + 2.74·17-s − 1.83·18-s + 2.32·19-s + 2.06·21-s + 4.29·22-s + 5.09·23-s + 4.27·24-s − 4.71·26-s − 5.63·27-s + 4.53·28-s + 3.18·29-s + 1.87·31-s − 5.55·32-s + 2.77·33-s + 6.30·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.856·3-s + 1.62·4-s + 1.38·6-s + 0.525·7-s + 1.01·8-s − 0.267·9-s + 0.565·11-s + 1.39·12-s − 0.570·13-s + 0.852·14-s + 0.0228·16-s + 0.666·17-s − 0.432·18-s + 0.532·19-s + 0.450·21-s + 0.916·22-s + 1.06·23-s + 0.871·24-s − 0.925·26-s − 1.08·27-s + 0.856·28-s + 0.591·29-s + 0.336·31-s − 0.981·32-s + 0.483·33-s + 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.269131781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.269131781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.887499671132764253904442163170, −8.970325851030386368802880282802, −8.134410799234057853115468369116, −7.20299455379139670122377880337, −6.36231824922654582982044224066, −5.26732116132311495300880107892, −4.72872534581682341526527593298, −3.49067135272860943510165284290, −3.00500540093619976153054343236, −1.78680273992652294318395634758,
1.78680273992652294318395634758, 3.00500540093619976153054343236, 3.49067135272860943510165284290, 4.72872534581682341526527593298, 5.26732116132311495300880107892, 6.36231824922654582982044224066, 7.20299455379139670122377880337, 8.134410799234057853115468369116, 8.970325851030386368802880282802, 9.887499671132764253904442163170
