| L(s) = 1 | − 4·7-s − 9-s + 2·11-s − 6·23-s − 5·25-s + 8·29-s + 14·37-s + 4·43-s + 9·49-s − 20·53-s + 4·63-s − 2·67-s − 2·71-s − 8·77-s − 4·79-s − 8·81-s − 2·99-s − 32·107-s + 4·109-s − 30·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s + 0.603·11-s − 1.25·23-s − 25-s + 1.48·29-s + 2.30·37-s + 0.609·43-s + 9/7·49-s − 2.74·53-s + 0.503·63-s − 0.244·67-s − 0.237·71-s − 0.911·77-s − 0.450·79-s − 8/9·81-s − 0.201·99-s − 3.09·107-s + 0.383·109-s − 2.82·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 157 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.133448943213416219981584405392, −8.311103393102123448321517998769, −7.972218436625198426991009353966, −7.54664461851502869881638278178, −6.73579323315780938126166238297, −6.40745311046018313656997838769, −6.07278572095843361061398546088, −5.64019672471834374599924808553, −4.74455528321738684868182201768, −4.14433958844327277498262470851, −3.74520522147862011695498567956, −2.89823787764158654024873948650, −2.55990091136445860728252253585, −1.35316500849431976367371795401, 0,
1.35316500849431976367371795401, 2.55990091136445860728252253585, 2.89823787764158654024873948650, 3.74520522147862011695498567956, 4.14433958844327277498262470851, 4.74455528321738684868182201768, 5.64019672471834374599924808553, 6.07278572095843361061398546088, 6.40745311046018313656997838769, 6.73579323315780938126166238297, 7.54664461851502869881638278178, 7.972218436625198426991009353966, 8.311103393102123448321517998769, 9.133448943213416219981584405392
