L(s) = 1 | − 2-s + 0.233·3-s + 4-s − 5-s − 0.233·6-s − 3.17·7-s − 8-s − 2.94·9-s + 10-s + 4.78·11-s + 0.233·12-s − 4.26·13-s + 3.17·14-s − 0.233·15-s + 16-s − 4.99·17-s + 2.94·18-s + 2.54·19-s − 20-s − 0.741·21-s − 4.78·22-s + 1.05·23-s − 0.233·24-s + 25-s + 4.26·26-s − 1.38·27-s − 3.17·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.134·3-s + 0.5·4-s − 0.447·5-s − 0.0953·6-s − 1.19·7-s − 0.353·8-s − 0.981·9-s + 0.316·10-s + 1.44·11-s + 0.0674·12-s − 1.18·13-s + 0.848·14-s − 0.0602·15-s + 0.250·16-s − 1.21·17-s + 0.694·18-s + 0.584·19-s − 0.223·20-s − 0.161·21-s − 1.02·22-s + 0.219·23-s − 0.0476·24-s + 0.200·25-s + 0.835·26-s − 0.267·27-s − 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6970320402\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6970320402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 223 | \( 1 - T \) |
good | 3 | \( 1 - 0.233T + 3T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 4.99T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 + 9.97T + 41T^{2} \) |
| 43 | \( 1 + 0.786T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 - 8.13T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 0.341T + 71T^{2} \) |
| 73 | \( 1 + 9.27T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−8.889355027294030088888009726790, −8.619483854698440219779444899210, −7.42060196916118700049868436345, −6.78954036701642299581152415878, −6.24662561565891814560324720861, −5.12341605625573393123901465570, −3.94139289880279331563675053146, −3.12298140948790461525935504855, −2.22368312517536055038337840149, −0.56969654288165106373775180523,
0.56969654288165106373775180523, 2.22368312517536055038337840149, 3.12298140948790461525935504855, 3.94139289880279331563675053146, 5.12341605625573393123901465570, 6.24662561565891814560324720861, 6.78954036701642299581152415878, 7.42060196916118700049868436345, 8.619483854698440219779444899210, 8.889355027294030088888009726790