| L(s) = 1 | − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s + 12-s − 7·16-s − 2·18-s + 4·19-s − 8·24-s + 25-s − 27-s − 4·29-s − 14·32-s − 36-s − 8·38-s + 20·41-s + 8·43-s + 7·48-s − 14·49-s − 2·50-s − 20·53-s + 2·54-s − 4·57-s + 8·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s + 0.288·12-s − 7/4·16-s − 0.471·18-s + 0.917·19-s − 1.63·24-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 2.47·32-s − 1/6·36-s − 1.29·38-s + 3.12·41-s + 1.21·43-s + 1.01·48-s − 2·49-s − 0.282·50-s − 2.74·53-s + 0.272·54-s − 0.529·57-s + 1.05·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.896447809747983381121393959041, −8.232011963291614168935511350860, −7.77071280677534797445852255244, −7.66488013441745380243230842523, −7.11524013136289051004798684067, −6.34145167556094658057083324143, −5.83679390833615760339265246745, −5.23920392624592057055772361749, −4.72109247762374245491183934060, −4.33195666036117235086666220277, −3.72492686467354363460541892516, −2.85380672919944569836723726469, −1.66341921070628603647910394136, −1.05029361783004892799606717843, 0,
1.05029361783004892799606717843, 1.66341921070628603647910394136, 2.85380672919944569836723726469, 3.72492686467354363460541892516, 4.33195666036117235086666220277, 4.72109247762374245491183934060, 5.23920392624592057055772361749, 5.83679390833615760339265246745, 6.34145167556094658057083324143, 7.11524013136289051004798684067, 7.66488013441745380243230842523, 7.77071280677534797445852255244, 8.232011963291614168935511350860, 8.896447809747983381121393959041
