| L(s) = 1 | − 4·4-s + 24·11-s + 16·16-s − 58·19-s − 204·29-s − 530·31-s + 480·41-s − 96·44-s + 325·49-s − 204·59-s − 206·61-s − 64·64-s − 1.16e3·71-s + 232·76-s − 346·79-s + 1.64e3·89-s − 2.40e3·101-s + 410·109-s + 816·116-s − 2.23e3·121-s + 2.12e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.657·11-s + 1/4·16-s − 0.700·19-s − 1.30·29-s − 3.07·31-s + 1.82·41-s − 0.328·44-s + 0.947·49-s − 0.450·59-s − 0.432·61-s − 1/8·64-s − 1.94·71-s + 0.350·76-s − 0.492·79-s + 1.95·89-s − 2.36·101-s + 0.360·109-s + 0.653·116-s − 1.67·121-s + 1.53·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2391036246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2391036246\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 325 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1894 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6050 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 265 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 97081 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 24325 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202462 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 p^{2} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 103 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 598822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 582 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 773809 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 173 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 p^{2} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 822 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1151305 T^{2} + p^{6} T^{4} \) |
| 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
−9.450923894528082337905424941450, −8.948228581000001923721119038019, −8.932819320433013685293452877503, −8.283448628614303342131233085043, −7.64437264043299810797003391126, −7.50698244159261628037807643379, −7.13813076925910733793089691943, −6.51364825569441487172515779019, −6.09869942739674825851724285854, −5.57771145859889737466631658844, −5.45048846859072180889708184532, −4.73276300285216113003933088414, −4.22515641553581824438072066493, −3.84114097815169696479660210201, −3.57831762654531185648570920684, −2.79981347286704247038586335848, −2.20331208157545179201644716195, −1.64004740752937379032158414104, −1.10393907569220871718466853577, −0.11983186337203598265607456313,
0.11983186337203598265607456313, 1.10393907569220871718466853577, 1.64004740752937379032158414104, 2.20331208157545179201644716195, 2.79981347286704247038586335848, 3.57831762654531185648570920684, 3.84114097815169696479660210201, 4.22515641553581824438072066493, 4.73276300285216113003933088414, 5.45048846859072180889708184532, 5.57771145859889737466631658844, 6.09869942739674825851724285854, 6.51364825569441487172515779019, 7.13813076925910733793089691943, 7.50698244159261628037807643379, 7.64437264043299810797003391126, 8.283448628614303342131233085043, 8.932819320433013685293452877503, 8.948228581000001923721119038019, 9.450923894528082337905424941450
