| L(s) = 1 | − 2-s + 2.26·3-s + 4-s − 3.49·5-s − 2.26·6-s + 4.18·7-s − 8-s + 2.11·9-s + 3.49·10-s − 3.90·11-s + 2.26·12-s + 2.34·13-s − 4.18·14-s − 7.90·15-s + 16-s + 3.24·17-s − 2.11·18-s + 0.106·19-s − 3.49·20-s + 9.46·21-s + 3.90·22-s + 5.38·23-s − 2.26·24-s + 7.20·25-s − 2.34·26-s − 1.99·27-s + 4.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.30·3-s + 0.5·4-s − 1.56·5-s − 0.923·6-s + 1.58·7-s − 0.353·8-s + 0.705·9-s + 1.10·10-s − 1.17·11-s + 0.652·12-s + 0.650·13-s − 1.11·14-s − 2.04·15-s + 0.250·16-s + 0.786·17-s − 0.498·18-s + 0.0243·19-s − 0.781·20-s + 2.06·21-s + 0.832·22-s + 1.12·23-s − 0.461·24-s + 1.44·25-s − 0.460·26-s − 0.384·27-s + 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.830949356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830949356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 - 0.106T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 + 0.525T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 7.53T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 0.621T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 4.50T + 89T^{2} \) |
| 97 | \( 1 - 5.58T + 97T^{2} \) |
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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
−8.519730532861270585439025229852, −8.151667803290905859118680354082, −7.57700413897962198656528445778, −7.30339202599981702920243468382, −5.66030170790492344683300297543, −4.73793797126597423064017789027, −3.83007605638792928528830363982, −3.06685809759105051012204971654, −2.13271118642722319306245825160, −0.901552772334582078932004515502,
0.901552772334582078932004515502, 2.13271118642722319306245825160, 3.06685809759105051012204971654, 3.83007605638792928528830363982, 4.73793797126597423064017789027, 5.66030170790492344683300297543, 7.30339202599981702920243468382, 7.57700413897962198656528445778, 8.151667803290905859118680354082, 8.519730532861270585439025229852
