| L(s) = 1 | − 2·2-s + 2·3-s − 4-s − 4·6-s + 8·8-s + 9-s − 11-s − 2·12-s − 7·16-s − 8·17-s − 2·18-s + 2·22-s + 16·24-s − 6·25-s − 4·27-s + 12·29-s + 20·31-s − 14·32-s − 2·33-s + 16·34-s − 36-s − 12·37-s − 8·41-s + 44-s − 14·48-s + 49-s + 12·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s + 2.82·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 7/4·16-s − 1.94·17-s − 0.471·18-s + 0.426·22-s + 3.26·24-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 3.59·31-s − 2.47·32-s − 0.348·33-s + 2.74·34-s − 1/6·36-s − 1.97·37-s − 1.24·41-s + 0.150·44-s − 2.02·48-s + 1/7·49-s + 1.69·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.339912824949481407935374648788, −8.184144826454463238531498469573, −7.67378178112872223010860918103, −6.94901622138190974050877028098, −6.70583207072298392541889211635, −6.09655823991520760248083514453, −5.04842931599159887263632672015, −4.94559945005359926667122742610, −4.28286203006294711588571475853, −3.97700068246488537440823967124, −3.17103530919462504679231815837, −2.48256179865490690419904115851, −1.91939450772296709675161614006, −1.01248464275202192351966482181, 0,
1.01248464275202192351966482181, 1.91939450772296709675161614006, 2.48256179865490690419904115851, 3.17103530919462504679231815837, 3.97700068246488537440823967124, 4.28286203006294711588571475853, 4.94559945005359926667122742610, 5.04842931599159887263632672015, 6.09655823991520760248083514453, 6.70583207072298392541889211635, 6.94901622138190974050877028098, 7.67378178112872223010860918103, 8.184144826454463238531498469573, 8.339912824949481407935374648788
