| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 8·7-s + 4·8-s + 9-s + 3·12-s + 16·14-s + 5·16-s + 2·18-s + 19-s + 8·21-s + 4·24-s − 10·25-s + 27-s + 24·28-s + 12·29-s + 6·32-s + 3·36-s + 2·38-s − 20·41-s + 16·42-s − 24·43-s + 5·48-s + 34·49-s − 20·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 3.02·7-s + 1.41·8-s + 1/3·9-s + 0.866·12-s + 4.27·14-s + 5/4·16-s + 0.471·18-s + 0.229·19-s + 1.74·21-s + 0.816·24-s − 2·25-s + 0.192·27-s + 4.53·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.324·38-s − 3.12·41-s + 2.46·42-s − 3.65·43-s + 0.721·48-s + 34/7·49-s − 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.360182093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.360182093\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.266538915527224263777195540327, −7.88755447490832193535409441723, −7.54953259660607698064814994541, −6.72056041985753664693998194219, −6.62468423904919083533508450719, −5.86279269185425347466345430550, −5.21883010312481367678358660757, −4.82624966188247411001253938074, −4.81736217668307600490535491911, −4.26615040210543801862743429615, −3.41899742877711894209698559306, −3.23599181183020074918802290096, −2.14151289942301727352812099402, −1.79376692971899737144429675124, −1.44213087893724210061243055494,
1.44213087893724210061243055494, 1.79376692971899737144429675124, 2.14151289942301727352812099402, 3.23599181183020074918802290096, 3.41899742877711894209698559306, 4.26615040210543801862743429615, 4.81736217668307600490535491911, 4.82624966188247411001253938074, 5.21883010312481367678358660757, 5.86279269185425347466345430550, 6.62468423904919083533508450719, 6.72056041985753664693998194219, 7.54953259660607698064814994541, 7.88755447490832193535409441723, 8.266538915527224263777195540327
