L(s) = 1 | + 3-s − 3·7-s + 9-s − 4·11-s + 7·13-s − 4·17-s + 19-s − 3·21-s − 8·23-s + 27-s + 3·31-s − 4·33-s − 2·37-s + 7·39-s − 6·41-s − 11·43-s + 6·47-s + 2·49-s − 4·51-s − 6·53-s + 57-s + 6·59-s − 61-s − 3·63-s − 15·67-s − 8·69-s − 6·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.20·11-s + 1.94·13-s − 0.970·17-s + 0.229·19-s − 0.654·21-s − 1.66·23-s + 0.192·27-s + 0.538·31-s − 0.696·33-s − 0.328·37-s + 1.12·39-s − 0.937·41-s − 1.67·43-s + 0.875·47-s + 2/7·49-s − 0.560·51-s − 0.824·53-s + 0.132·57-s + 0.781·59-s − 0.128·61-s − 0.377·63-s − 1.83·67-s − 0.963·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.445548395842484462406119273988, −8.096621371754884095752066096889, −6.95611886681030858986820779992, −6.30181751747046249588805643143, −5.59442823218209652422679327177, −4.36405852597219641768616689481, −3.54305188871072929959679365713, −2.84018741911282522171877844136, −1.69836775632297947459606316502, 0,
1.69836775632297947459606316502, 2.84018741911282522171877844136, 3.54305188871072929959679365713, 4.36405852597219641768616689481, 5.59442823218209652422679327177, 6.30181751747046249588805643143, 6.95611886681030858986820779992, 8.096621371754884095752066096889, 8.445548395842484462406119273988