| L(s) = 1 | + 3·3-s + 7-s + 6·9-s − 2·11-s − 13-s − 3·17-s + 19-s + 3·21-s − 3·23-s + 9·27-s − 3·29-s + 8·31-s − 6·33-s − 10·37-s − 3·39-s − 12·41-s + 8·43-s + 8·47-s − 6·49-s − 9·51-s − 9·53-s + 3·57-s + 5·59-s − 10·61-s + 6·63-s + 7·67-s − 9·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s − 0.603·11-s − 0.277·13-s − 0.727·17-s + 0.229·19-s + 0.654·21-s − 0.625·23-s + 1.73·27-s − 0.557·29-s + 1.43·31-s − 1.04·33-s − 1.64·37-s − 0.480·39-s − 1.87·41-s + 1.21·43-s + 1.16·47-s − 6/7·49-s − 1.26·51-s − 1.23·53-s + 0.397·57-s + 0.650·59-s − 1.28·61-s + 0.755·63-s + 0.855·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.32728564883435, −14.85257127539701, −14.14556193016873, −13.99088482193569, −13.34333929255879, −13.03918934740744, −12.20614883885938, −11.86786129262323, −10.93714032186302, −10.42902674959462, −9.906569163959297, −9.334727321974765, −8.829407629510560, −8.177608609724821, −8.038378548244629, −7.217346669414511, −6.860738122927036, −5.949675705768035, −5.121720329115088, −4.515553765855351, −3.934196479432109, −3.183710433725236, −2.679327810533829, −1.994877884106297, −1.416502635537563, 0,
1.416502635537563, 1.994877884106297, 2.679327810533829, 3.183710433725236, 3.934196479432109, 4.515553765855351, 5.121720329115088, 5.949675705768035, 6.860738122927036, 7.217346669414511, 8.038378548244629, 8.177608609724821, 8.829407629510560, 9.334727321974765, 9.906569163959297, 10.42902674959462, 10.93714032186302, 11.86786129262323, 12.20614883885938, 13.03918934740744, 13.34333929255879, 13.99088482193569, 14.14556193016873, 14.85257127539701, 15.32728564883435
