| L(s) = 1 | − 0.414·2-s + 3·3-s − 7.82·4-s − 0.100·5-s − 1.24·6-s + 6.55·8-s + 9·9-s + 0.0416·10-s − 43.9·11-s − 23.4·12-s − 16.6·13-s − 0.301·15-s + 59.9·16-s − 121.·17-s − 3.72·18-s − 127.·19-s + 0.786·20-s + 18.2·22-s + 53.5·23-s + 19.6·24-s − 124.·25-s + 6.89·26-s + 27·27-s + 235.·29-s + 0.124·30-s − 18.7·31-s − 77.2·32-s + ⋯ |
| L(s) = 1 | − 0.146·2-s + 0.577·3-s − 0.978·4-s − 0.00898·5-s − 0.0845·6-s + 0.289·8-s + 0.333·9-s + 0.00131·10-s − 1.20·11-s − 0.564·12-s − 0.355·13-s − 0.00519·15-s + 0.936·16-s − 1.73·17-s − 0.0488·18-s − 1.53·19-s + 0.00879·20-s + 0.176·22-s + 0.485·23-s + 0.167·24-s − 0.999·25-s + 0.0520·26-s + 0.192·27-s + 1.50·29-s + 0.000760·30-s − 0.108·31-s − 0.426·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 8T^{2} \) |
| 5 | \( 1 + 0.100T + 125T^{2} \) |
| 11 | \( 1 + 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 11.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 12.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 805.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 394.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−12.38467057661612966517169816014, −10.77093414912933788232081684469, −9.980904315594220259933380047603, −8.750325995407371482527423385261, −8.230850937195979633243153112089, −6.82051341125267135912392837140, −5.12565370016064279247676634638, −4.08766976519234082274965571733, −2.36582910168422816302472070262, 0,
2.36582910168422816302472070262, 4.08766976519234082274965571733, 5.12565370016064279247676634638, 6.82051341125267135912392837140, 8.230850937195979633243153112089, 8.750325995407371482527423385261, 9.980904315594220259933380047603, 10.77093414912933788232081684469, 12.38467057661612966517169816014
