| L(s) = 1 | − 2.53·2-s + 3-s + 4.42·4-s − 3.70·5-s − 2.53·6-s + 0.957·7-s − 6.14·8-s + 9-s + 9.37·10-s − 11-s + 4.42·12-s − 2.42·14-s − 3.70·15-s + 6.71·16-s − 2.05·17-s − 2.53·18-s − 7.67·19-s − 16.3·20-s + 0.957·21-s + 2.53·22-s − 4.20·23-s − 6.14·24-s + 8.69·25-s + 27-s + 4.23·28-s + 1.97·29-s + 9.37·30-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.21·4-s − 1.65·5-s − 1.03·6-s + 0.361·7-s − 2.17·8-s + 0.333·9-s + 2.96·10-s − 0.301·11-s + 1.27·12-s − 0.648·14-s − 0.955·15-s + 1.67·16-s − 0.498·17-s − 0.597·18-s − 1.76·19-s − 3.66·20-s + 0.208·21-s + 0.540·22-s − 0.877·23-s − 1.25·24-s + 1.73·25-s + 0.192·27-s + 0.800·28-s + 0.367·29-s + 1.71·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 0.957T + 7T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.183T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 + 5.24T + 89T^{2} \) |
| 97 | \( 1 + 9.82T + 97T^{2} \) |
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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
−8.067894937790211162361901598580, −7.49433853490838232854569938283, −6.78852857886727599656139967006, −6.09776374971821887927638428953, −4.46763979084550424373925699292, −4.13371792720002971519621422540, −2.84095805239973846171021404679, −2.25217355790317773516925233489, −0.988714589671065127402674126138, 0,
0.988714589671065127402674126138, 2.25217355790317773516925233489, 2.84095805239973846171021404679, 4.13371792720002971519621422540, 4.46763979084550424373925699292, 6.09776374971821887927638428953, 6.78852857886727599656139967006, 7.49433853490838232854569938283, 8.067894937790211162361901598580
