| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s − 10·25-s − 16·29-s − 6·32-s + 4·37-s + 8·43-s + 6·44-s + 20·50-s − 4·53-s + 32·58-s + 7·64-s + 16·67-s + 4·71-s − 8·74-s + 32·79-s − 16·86-s − 8·88-s − 30·100-s + 8·106-s + 4·107-s + 32·109-s + 4·113-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 − 2·25-s − 2.97·29-s − 1.06·32-s + 0.657·37-s + 1.21·43-s + 0.904·44-s + 2.82·50-s − 0.549·53-s + 4.20·58-s + 7/8·64-s + 1.95·67-s + 0.474·71-s − 0.929·74-s + 3.60·79-s − 1.72·86-s − 0.852·88-s − 3·100-s + 0.777·106-s + 0.386·107-s + 3.06·109-s + 0.376·113-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
| \(L(1)\) |
\(\approx\) |
\(1.264545331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264545331\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70567297607640320107453151363, −7.67297754577719580753689022662, −7.19499782591586439086948792633, −7.17344089085263677721691624186, −6.38930320612238921916073088198, −6.31958190631041893542918077877, −5.83314430942536371656977827116, −5.77659684824194118493288477656, −5.12646707043840207919541811060, −4.90003667058348191766168852249, −4.19412279215687235815424572631, −3.85047590392538865886136671199, −3.55765579696241979532889529487, −3.35111869558200821282547366724, −2.44328318858504631327385476140, −2.27412780504869153407906102209, −1.85904967420795790914494614938, −1.53645055161741616432548645839, −0.72988484977764104177797296407, −0.43800977047981705021137114939,
0.43800977047981705021137114939, 0.72988484977764104177797296407, 1.53645055161741616432548645839, 1.85904967420795790914494614938, 2.27412780504869153407906102209, 2.44328318858504631327385476140, 3.35111869558200821282547366724, 3.55765579696241979532889529487, 3.85047590392538865886136671199, 4.19412279215687235815424572631, 4.90003667058348191766168852249, 5.12646707043840207919541811060, 5.77659684824194118493288477656, 5.83314430942536371656977827116, 6.31958190631041893542918077877, 6.38930320612238921916073088198, 7.17344089085263677721691624186, 7.19499782591586439086948792633, 7.67297754577719580753689022662, 7.70567297607640320107453151363
