| L(s) = 1 | + 3-s − 2·5-s + 4·7-s − 2·9-s − 3·11-s − 2·15-s − 3·17-s − 5·19-s + 4·21-s − 25-s − 5·27-s − 6·29-s + 6·31-s − 3·33-s − 8·35-s + 4·37-s − 12·41-s + 9·43-s + 4·45-s + 6·47-s + 9·49-s − 3·51-s + 11·53-s + 6·55-s − 5·57-s + 7·59-s − 8·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s − 2/3·9-s − 0.904·11-s − 0.516·15-s − 0.727·17-s − 1.14·19-s + 0.872·21-s − 1/5·25-s − 0.962·27-s − 1.11·29-s + 1.07·31-s − 0.522·33-s − 1.35·35-s + 0.657·37-s − 1.87·41-s + 1.37·43-s + 0.596·45-s + 0.875·47-s + 9/7·49-s − 0.420·51-s + 1.51·53-s + 0.809·55-s − 0.662·57-s + 0.911·59-s − 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 6571 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.91140960877719, −13.60747081909884, −13.00257291667240, −12.53125263524837, −11.76439260068075, −11.45678731989396, −11.24265261146645, −10.45807044515926, −10.29747216480054, −9.284395047633666, −8.824472106715445, −8.299339994595314, −8.167630005905517, −7.580043691694828, −7.197441211761739, −6.386802829348182, −5.719927624725069, −5.200074813316136, −4.670595556984474, −4.036000132427654, −3.751706725912374, −2.760022842898081, −2.304804527504985, −1.864555387109934, −0.7806973643122505, 0,
0.7806973643122505, 1.864555387109934, 2.304804527504985, 2.760022842898081, 3.751706725912374, 4.036000132427654, 4.670595556984474, 5.200074813316136, 5.719927624725069, 6.386802829348182, 7.197441211761739, 7.580043691694828, 8.167630005905517, 8.299339994595314, 8.824472106715445, 9.284395047633666, 10.29747216480054, 10.45807044515926, 11.24265261146645, 11.45678731989396, 11.76439260068075, 12.53125263524837, 13.00257291667240, 13.60747081909884, 13.91140960877719
