| 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
| \(L(1)\) |
\(\approx\) |
\(10.53787776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.53787776\) |
| \(L(\frac{3}{2})\) |
|
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
−7.68818697631310143305033485340, −7.60583815744645128990979308144, −6.92523528827935007794511073019, −6.81931202910993732601588500771, −6.26524804403649605124419349039, −6.21852055343393834301385029747, −5.69026198541919167325047066703, −5.52833031378933305037005008648, −4.93158922715432313029288632599, −4.84677321962553711672209240861, −4.24431399712011054340352179156, −4.24209284410225640670333308691, −3.51125655789984527987765712501, −3.44946515775012672982997885449, −2.77942593533747286287027207948, −2.56749623835425808562705659034, −2.06084691522551585932591173944, −1.83798536697278375751835959859, −0.833541153332378404235166859028, −0.70267793648830110779998677227,
0.70267793648830110779998677227, 0.833541153332378404235166859028, 1.83798536697278375751835959859, 2.06084691522551585932591173944, 2.56749623835425808562705659034, 2.77942593533747286287027207948, 3.44946515775012672982997885449, 3.51125655789984527987765712501, 4.24209284410225640670333308691, 4.24431399712011054340352179156, 4.84677321962553711672209240861, 4.93158922715432313029288632599, 5.52833031378933305037005008648, 5.69026198541919167325047066703, 6.21852055343393834301385029747, 6.26524804403649605124419349039, 6.81931202910993732601588500771, 6.92523528827935007794511073019, 7.60583815744645128990979308144, 7.68818697631310143305033485340
