L(s) = 1 | + 4·2-s + 3-s + 8·4-s + 4·6-s − 4·7-s + 8·8-s − 2·9-s + 8·12-s − 16·14-s − 4·16-s − 8·18-s − 4·21-s + 8·24-s − 9·25-s − 5·27-s − 32·28-s − 32·32-s − 16·36-s + 16·41-s − 16·42-s − 12·43-s − 4·48-s − 2·49-s − 36·50-s + 12·53-s − 20·54-s − 32·56-s + ⋯ |
L(s) = 1 | + 2.82·2-s + 0.577·3-s + 4·4-s + 1.63·6-s − 1.51·7-s + 2.82·8-s − 2/3·9-s + 2.30·12-s − 4.27·14-s − 16-s − 1.88·18-s − 0.872·21-s + 1.63·24-s − 9/5·25-s − 0.962·27-s − 6.04·28-s − 5.65·32-s − 8/3·36-s + 2.49·41-s − 2.46·42-s − 1.82·43-s − 0.577·48-s − 2/7·49-s − 5.09·50-s + 1.64·53-s − 2.72·54-s − 4.27·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.228854358433494441874279371750, −7.991382198601661869330644512838, −7.08610298939652805070973785571, −6.66503987304844816686807720890, −6.44705964567155032403967809781, −5.66407660024727617127120629681, −5.65921461221479271613429705311, −5.19096686691774806327844897047, −4.19085115174357929506261869806, −4.09468272193076097429923581775, −3.55069629742260821889174578377, −3.09000916592887094247460325417, −2.62758010025505468758993615534, −2.12638900923174362608185387010, 0,
2.12638900923174362608185387010, 2.62758010025505468758993615534, 3.09000916592887094247460325417, 3.55069629742260821889174578377, 4.09468272193076097429923581775, 4.19085115174357929506261869806, 5.19096686691774806327844897047, 5.65921461221479271613429705311, 5.66407660024727617127120629681, 6.44705964567155032403967809781, 6.66503987304844816686807720890, 7.08610298939652805070973785571, 7.991382198601661869330644512838, 8.228854358433494441874279371750