| L(s) = 1 | + 3·5-s − 3·11-s − 2·13-s + 6·17-s + 2·19-s − 6·23-s + 4·25-s − 9·29-s − 7·31-s − 10·37-s + 4·43-s − 12·47-s + 3·53-s − 9·55-s + 3·59-s + 4·61-s − 6·65-s − 2·67-s − 2·73-s − 5·79-s − 9·83-s + 18·85-s − 6·89-s + 6·95-s + 13·97-s + 6·101-s − 16·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.904·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 4/5·25-s − 1.67·29-s − 1.25·31-s − 1.64·37-s + 0.609·43-s − 1.75·47-s + 0.412·53-s − 1.21·55-s + 0.390·59-s + 0.512·61-s − 0.744·65-s − 0.244·67-s − 0.234·73-s − 0.562·79-s − 0.987·83-s + 1.95·85-s − 0.635·89-s + 0.615·95-s + 1.31·97-s + 0.597·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.55897061209615858126116845368, −6.97910161712974459656407065808, −5.90823736302778728203678186086, −5.52219615915251146899126346033, −5.09851582433884081701270571273, −3.85477814317456269292247844951, −3.08421331729666561174533371760, −2.12501224527283536650390996746, −1.55872607484065073233825464189, 0,
1.55872607484065073233825464189, 2.12501224527283536650390996746, 3.08421331729666561174533371760, 3.85477814317456269292247844951, 5.09851582433884081701270571273, 5.52219615915251146899126346033, 5.90823736302778728203678186086, 6.97910161712974459656407065808, 7.55897061209615858126116845368
