| L(s) = 1 | + 3.23·2-s − 3·3-s + 2.47·4-s + 13.5·5-s − 9.70·6-s − 1.42·7-s − 17.8·8-s + 9·9-s + 43.9·10-s − 54.5·11-s − 7.42·12-s − 4.62·14-s − 40.7·15-s − 77.6·16-s − 114.·17-s + 29.1·18-s + 104.·19-s + 33.6·20-s + 4.28·21-s − 176.·22-s − 64.5·23-s + 53.6·24-s + 59.4·25-s − 27·27-s − 3.53·28-s − 60.8·29-s − 131.·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s + 1.21·5-s − 0.660·6-s − 0.0771·7-s − 0.790·8-s + 0.333·9-s + 1.39·10-s − 1.49·11-s − 0.178·12-s − 0.0883·14-s − 0.701·15-s − 1.21·16-s − 1.63·17-s + 0.381·18-s + 1.26·19-s + 0.375·20-s + 0.0445·21-s − 1.71·22-s − 0.585·23-s + 0.456·24-s + 0.475·25-s − 0.192·27-s − 0.0238·28-s − 0.389·29-s − 0.802·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.23T + 8T^{2} \) |
| 5 | \( 1 - 13.5T + 125T^{2} \) |
| 7 | \( 1 + 1.42T + 343T^{2} \) |
| 11 | \( 1 + 54.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 639.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 819.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 574.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 365.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 965.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 580.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 20.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−10.09194148984188661447631708456, −9.442219237143777839621647793176, −8.223372088245044790769180914709, −6.82485241645929945291419809586, −6.01974616973875791147863171752, −5.26223134325318024929002492377, −4.63873806686654011397416089362, −3.12702505547620047103934775624, −2.03742627509707335826509127344, 0,
2.03742627509707335826509127344, 3.12702505547620047103934775624, 4.63873806686654011397416089362, 5.26223134325318024929002492377, 6.01974616973875791147863171752, 6.82485241645929945291419809586, 8.223372088245044790769180914709, 9.442219237143777839621647793176, 10.09194148984188661447631708456
