| L(s) = 1 | + 3-s + 2·5-s − 4·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 6·17-s − 4·19-s − 4·21-s − 25-s + 27-s + 2·29-s + 4·31-s + 4·33-s − 8·35-s − 2·37-s − 2·39-s + 2·41-s + 4·43-s + 2·45-s + 8·47-s + 9·49-s − 6·51-s + 10·53-s + 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 1.35·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171420084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171420084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.80073173066515229603635969273, −13.14140490363893974463899954609, −12.10541458487679021094810242454, −10.44913304188171401892835914004, −9.488699237661810053920246470193, −8.839886292016704738441339481672, −6.91469103626909406223255043804, −6.15267013015603292112906972931, −4.10625798014363516984369255368, −2.44821729992400328434125228202,
2.44821729992400328434125228202, 4.10625798014363516984369255368, 6.15267013015603292112906972931, 6.91469103626909406223255043804, 8.839886292016704738441339481672, 9.488699237661810053920246470193, 10.44913304188171401892835914004, 12.10541458487679021094810242454, 13.14140490363893974463899954609, 13.80073173066515229603635969273
