| L(s) = 1 | − 0.844·2-s − 3-s − 1.28·4-s + 1.91·5-s + 0.844·6-s + 2.77·8-s + 9-s − 1.61·10-s + 2.68·11-s + 1.28·12-s − 1.87·13-s − 1.91·15-s + 0.231·16-s − 6.09·17-s − 0.844·18-s − 7.23·19-s − 2.46·20-s − 2.26·22-s − 3.42·23-s − 2.77·24-s − 1.34·25-s + 1.58·26-s − 27-s + 10.5·29-s + 1.61·30-s − 7.41·31-s − 5.74·32-s + ⋯ |
| L(s) = 1 | − 0.596·2-s − 0.577·3-s − 0.643·4-s + 0.855·5-s + 0.344·6-s + 0.981·8-s + 0.333·9-s − 0.510·10-s + 0.808·11-s + 0.371·12-s − 0.521·13-s − 0.493·15-s + 0.0579·16-s − 1.47·17-s − 0.198·18-s − 1.66·19-s − 0.550·20-s − 0.482·22-s − 0.714·23-s − 0.566·24-s − 0.268·25-s + 0.311·26-s − 0.192·27-s + 1.95·29-s + 0.294·30-s − 1.33·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8017389487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8017389487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.844T + 2T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 + 4.04T + 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 - 7.18T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 - 4.66T + 97T^{2} \) |
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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
−8.367675022534537220705240932335, −7.27778858211712628665322570985, −6.64700166140725552282599855900, −6.03027272100334651283305159387, −5.24910387821779513181614456156, −4.32685301629724765180927102096, −4.06587811593515271762121463329, −2.39328218944516950971855378507, −1.74531858948615903153832261885, −0.53401906489958562517152301149,
0.53401906489958562517152301149, 1.74531858948615903153832261885, 2.39328218944516950971855378507, 4.06587811593515271762121463329, 4.32685301629724765180927102096, 5.24910387821779513181614456156, 6.03027272100334651283305159387, 6.64700166140725552282599855900, 7.27778858211712628665322570985, 8.367675022534537220705240932335
