| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s − 2·25-s + 16·29-s − 6·32-s + 12·37-s + 8·43-s + 6·44-s + 4·50-s − 12·53-s − 32·58-s + 7·64-s − 8·67-s − 12·71-s − 24·74-s + 24·79-s − 16·86-s − 8·88-s − 6·100-s + 24·106-s + 12·107-s − 32·109-s + 28·113-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s + 5/4·16-s − 0.852·22-s − 2/5·25-s + 2.97·29-s − 1.06·32-s + 1.97·37-s + 1.21·43-s + 0.904·44-s + 0.565·50-s − 1.64·53-s − 4.20·58-s + 7/8·64-s − 0.977·67-s − 1.42·71-s − 2.78·74-s + 2.70·79-s − 1.72·86-s − 0.852·88-s − 3/5·100-s + 2.33·106-s + 1.16·107-s − 3.06·109-s + 2.63·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.102729754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.102729754\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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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
−7.83537852237884681634256797932, −7.79394677271858966291240320679, −7.15135258304350382991587790599, −6.93988656316684509968173879860, −6.46505940166794844496082829937, −6.38101463117460336181610633857, −5.87160508176530926687417747915, −5.80418408996490465920566023651, −5.06006049345489177531043577097, −4.72885407415905713250781579951, −4.27712163915908655512124860579, −4.16645886722254976304676497846, −3.35612778805216071383875098354, −3.11690799875524098890339725048, −2.68234948616775023940103290074, −2.38203924329561205200159889109, −1.75024583276606341035035693639, −1.40557896966829901778368330167, −0.72042165436700170120877742735, −0.61773261242621826508170215345,
0.61773261242621826508170215345, 0.72042165436700170120877742735, 1.40557896966829901778368330167, 1.75024583276606341035035693639, 2.38203924329561205200159889109, 2.68234948616775023940103290074, 3.11690799875524098890339725048, 3.35612778805216071383875098354, 4.16645886722254976304676497846, 4.27712163915908655512124860579, 4.72885407415905713250781579951, 5.06006049345489177531043577097, 5.80418408996490465920566023651, 5.87160508176530926687417747915, 6.38101463117460336181610633857, 6.46505940166794844496082829937, 6.93988656316684509968173879860, 7.15135258304350382991587790599, 7.79394677271858966291240320679, 7.83537852237884681634256797932
