| L(s) = 1 | + 0.404·2-s − 3-s − 1.83·4-s + 3.62·5-s − 0.404·6-s − 1.55·8-s + 9-s + 1.46·10-s − 5.64·11-s + 1.83·12-s − 4.03·13-s − 3.62·15-s + 3.04·16-s − 2.00·17-s + 0.404·18-s + 1.50·19-s − 6.66·20-s − 2.28·22-s + 7.05·23-s + 1.55·24-s + 8.15·25-s − 1.63·26-s − 27-s + 5.14·29-s − 1.46·30-s + 3.25·31-s + 4.33·32-s + ⋯ |
| L(s) = 1 | + 0.285·2-s − 0.577·3-s − 0.918·4-s + 1.62·5-s − 0.165·6-s − 0.548·8-s + 0.333·9-s + 0.463·10-s − 1.70·11-s + 0.530·12-s − 1.11·13-s − 0.936·15-s + 0.761·16-s − 0.487·17-s + 0.0952·18-s + 0.346·19-s − 1.48·20-s − 0.486·22-s + 1.47·23-s + 0.316·24-s + 1.63·25-s − 0.320·26-s − 0.192·27-s + 0.955·29-s − 0.267·30-s + 0.585·31-s + 0.766·32-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.404T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 - 0.612T + 37T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 + 0.256T + 47T^{2} \) |
| 53 | \( 1 + 7.87T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 2.81T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 9.74T + 89T^{2} \) |
| 97 | \( 1 - 8.22T + 97T^{2} \) |
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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
−7.69858410923844914764087571446, −6.83964858346375320599950716170, −6.10158662170666606959102158733, −5.30179283780469959887514657184, −5.09216050239369514839986300477, −4.47243480922702054465625852412, −2.94327415013231547761038911636, −2.53527546317837936630467013528, −1.24157544235774194989864210816, 0,
1.24157544235774194989864210816, 2.53527546317837936630467013528, 2.94327415013231547761038911636, 4.47243480922702054465625852412, 5.09216050239369514839986300477, 5.30179283780469959887514657184, 6.10158662170666606959102158733, 6.83964858346375320599950716170, 7.69858410923844914764087571446
