| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 6·11-s − 2·13-s + 16-s + 6·17-s + 7·19-s − 3·20-s + 6·22-s − 3·23-s + 4·25-s − 2·26-s − 6·29-s − 2·31-s + 32-s + 6·34-s + 2·37-s + 7·38-s − 3·40-s + 2·43-s + 6·44-s − 3·46-s + 4·50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 1.80·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 1.60·19-s − 0.670·20-s + 1.27·22-s − 0.625·23-s + 4/5·25-s − 0.392·26-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 1.13·38-s − 0.474·40-s + 0.304·43-s + 0.904·44-s − 0.442·46-s + 0.565·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.948191922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.948191922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.52615826214804217802768055243, −7.37366030853115071117514084487, −6.43237543258924286821522381914, −5.69590077176605674126774733821, −4.95489136451730675655372502871, −4.10633662675451992991393928266, −3.61372912707775721400155498287, −3.12413911599566555674994589114, −1.74168813292646473234241379022, −0.801100700285570341020354788436,
0.801100700285570341020354788436, 1.74168813292646473234241379022, 3.12413911599566555674994589114, 3.61372912707775721400155498287, 4.10633662675451992991393928266, 4.95489136451730675655372502871, 5.69590077176605674126774733821, 6.43237543258924286821522381914, 7.37366030853115071117514084487, 7.52615826214804217802768055243
