| L(s) = 1 | + 3-s − 2·7-s + 9-s + 4·11-s + 13-s + 2·19-s − 2·21-s + 2·23-s + 27-s + 4·29-s + 4·31-s + 4·33-s + 2·37-s + 39-s − 6·41-s + 4·43-s − 8·47-s − 3·49-s + 2·53-s + 2·57-s + 4·59-s − 2·61-s − 2·63-s − 8·67-s + 2·69-s + 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.458·19-s − 0.436·21-s + 0.417·23-s + 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s + 0.240·69-s + 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.699401892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.699401892\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.970887846116897845337911764056, −7.00602061281196743622001027859, −6.60925303064095719357376751039, −5.94627902094419587493385407810, −4.93678056529372727717078038163, −4.17706540036201829301436535489, −3.41731248257339669585809020486, −2.88127452579023111506453617323, −1.76204806578458563268563154525, −0.825290408356212376382984222921,
0.825290408356212376382984222921, 1.76204806578458563268563154525, 2.88127452579023111506453617323, 3.41731248257339669585809020486, 4.17706540036201829301436535489, 4.93678056529372727717078038163, 5.94627902094419587493385407810, 6.60925303064095719357376751039, 7.00602061281196743622001027859, 7.970887846116897845337911764056
