| L(s) = 1 | + 3-s + 2·7-s − 2·9-s − 4·11-s − 13-s − 4·19-s + 2·21-s + 23-s − 5·27-s − 7·29-s − 7·31-s − 4·33-s + 4·37-s − 39-s + 3·41-s − 6·43-s + 13·47-s − 3·49-s − 10·53-s − 4·57-s − 8·59-s − 4·63-s − 8·67-s + 69-s + 13·71-s − 11·73-s − 8·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s − 2/3·9-s − 1.20·11-s − 0.277·13-s − 0.917·19-s + 0.436·21-s + 0.208·23-s − 0.962·27-s − 1.29·29-s − 1.25·31-s − 0.696·33-s + 0.657·37-s − 0.160·39-s + 0.468·41-s − 0.914·43-s + 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.529·57-s − 1.04·59-s − 0.503·63-s − 0.977·67-s + 0.120·69-s + 1.54·71-s − 1.28·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.640761851859682184053942685631, −7.75671950520096891573647096692, −7.50003401681504724026745936327, −6.15921832067481142134696113819, −5.41646555025434615264403797688, −4.65662052732943133220446353251, −3.59422562930134537932914640730, −2.61635712140400996842197655793, −1.86021166262617436947610594676, 0,
1.86021166262617436947610594676, 2.61635712140400996842197655793, 3.59422562930134537932914640730, 4.65662052732943133220446353251, 5.41646555025434615264403797688, 6.15921832067481142134696113819, 7.50003401681504724026745936327, 7.75671950520096891573647096692, 8.640761851859682184053942685631
