| L(s) = 1 | + 2.08·2-s − 3-s + 2.33·4-s + 1.37·5-s − 2.08·6-s + 0.703·8-s + 9-s + 2.87·10-s − 1.74·11-s − 2.33·12-s − 0.365·13-s − 1.37·15-s − 3.21·16-s − 8.01·17-s + 2.08·18-s − 5.54·19-s + 3.22·20-s − 3.63·22-s + 23-s − 0.703·24-s − 3.09·25-s − 0.761·26-s − 27-s + 7.33·29-s − 2.87·30-s − 4.71·31-s − 8.09·32-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.577·3-s + 1.16·4-s + 0.616·5-s − 0.850·6-s + 0.248·8-s + 0.333·9-s + 0.908·10-s − 0.526·11-s − 0.674·12-s − 0.101·13-s − 0.356·15-s − 0.802·16-s − 1.94·17-s + 0.490·18-s − 1.27·19-s + 0.721·20-s − 0.774·22-s + 0.208·23-s − 0.143·24-s − 0.619·25-s − 0.149·26-s − 0.192·27-s + 1.36·29-s − 0.524·30-s − 0.847·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 + 0.365T + 13T^{2} \) |
| 17 | \( 1 + 8.01T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 + 4.36T + 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 1.18T + 59T^{2} \) |
| 61 | \( 1 - 3.66T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 0.114T + 97T^{2} \) |
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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
−8.199313785409431532682798390127, −7.00785149575004783626815497376, −6.36117486073228025993758024996, −6.03482761047054895770165628631, −4.92351191810153062431118745760, −4.67427494068115249679561885040, −3.74886845654443143788209730877, −2.59369197473034699007243650489, −1.95133603143119443259183780877, 0,
1.95133603143119443259183780877, 2.59369197473034699007243650489, 3.74886845654443143788209730877, 4.67427494068115249679561885040, 4.92351191810153062431118745760, 6.03482761047054895770165628631, 6.36117486073228025993758024996, 7.00785149575004783626815497376, 8.199313785409431532682798390127
