| L(s) = 1 | + 2.77·2-s + 3-s + 5.67·4-s − 3.19·5-s + 2.77·6-s − 4.02·7-s + 10.1·8-s + 9-s − 8.84·10-s + 2.98·11-s + 5.67·12-s + 13-s − 11.1·14-s − 3.19·15-s + 16.8·16-s − 2.23·17-s + 2.77·18-s + 4.64·19-s − 18.1·20-s − 4.02·21-s + 8.27·22-s + 8.18·23-s + 10.1·24-s + 5.20·25-s + 2.77·26-s + 27-s − 22.8·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.577·3-s + 2.83·4-s − 1.42·5-s + 1.13·6-s − 1.52·7-s + 3.60·8-s + 0.333·9-s − 2.79·10-s + 0.900·11-s + 1.63·12-s + 0.277·13-s − 2.97·14-s − 0.824·15-s + 4.21·16-s − 0.541·17-s + 0.653·18-s + 1.06·19-s − 4.05·20-s − 0.878·21-s + 1.76·22-s + 1.70·23-s + 2.07·24-s + 1.04·25-s + 0.543·26-s + 0.192·27-s − 4.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.755108376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.755108376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 - 4.91T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 + 9.52T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 + 6.52T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 + 9.52T + 61T^{2} \) |
| 67 | \( 1 - 4.02T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.130379265864658613921778304615, −7.35200534363217542498630372223, −6.73290923851984572574214561054, −6.39570854618806884788575729739, −5.25026486008489911822263751552, −4.36312371597764512771346777747, −3.86519377550434548235471160248, −3.10613709260938241930935251243, −2.84706159968011100004254738951, −1.15496722177387414835317188707,
1.15496722177387414835317188707, 2.84706159968011100004254738951, 3.10613709260938241930935251243, 3.86519377550434548235471160248, 4.36312371597764512771346777747, 5.25026486008489911822263751552, 6.39570854618806884788575729739, 6.73290923851984572574214561054, 7.35200534363217542498630372223, 8.130379265864658613921778304615
