| L(s) = 1 | + 3-s − 1.61·7-s − 2·9-s + 3·11-s − 5.47·13-s − 1.14·17-s + 7.23·19-s − 1.61·21-s + 1.85·23-s − 5·27-s + 6.70·29-s + 5.76·31-s + 3·33-s + 6.09·37-s − 5.47·39-s − 9.70·41-s − 12.0·43-s − 3·47-s − 4.38·49-s − 1.14·51-s − 12.7·53-s + 7.23·57-s + 6.70·59-s − 5.76·61-s + 3.23·63-s + 2.52·67-s + 1.85·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.611·7-s − 0.666·9-s + 0.904·11-s − 1.51·13-s − 0.277·17-s + 1.66·19-s − 0.353·21-s + 0.386·23-s − 0.962·27-s + 1.24·29-s + 1.03·31-s + 0.522·33-s + 1.00·37-s − 0.876·39-s − 1.51·41-s − 1.84·43-s − 0.437·47-s − 0.625·49-s − 0.160·51-s − 1.74·53-s + 0.958·57-s + 0.873·59-s − 0.737·61-s + 0.407·63-s + 0.308·67-s + 0.223·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−7.30727658876430549595620789057, −6.65398749656480104441626494921, −6.13704077925271133933761089075, −5.02849418284050656553198489972, −4.74036531769910119622627031153, −3.45604154247264052264501046363, −3.11281789588813753710563788295, −2.36149371884352804546634187882, −1.24475709255520853155903449350, 0,
1.24475709255520853155903449350, 2.36149371884352804546634187882, 3.11281789588813753710563788295, 3.45604154247264052264501046363, 4.74036531769910119622627031153, 5.02849418284050656553198489972, 6.13704077925271133933761089075, 6.65398749656480104441626494921, 7.30727658876430549595620789057
