L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 4·11-s − 12-s − 2·13-s + 16-s − 2·18-s + 4·22-s − 2·23-s − 24-s + 6·25-s − 2·26-s + 5·27-s + 32-s − 4·33-s − 2·36-s − 4·37-s + 2·39-s + 4·44-s − 2·46-s + 16·47-s − 48-s − 5·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.471·18-s + 0.852·22-s − 0.417·23-s − 0.204·24-s + 6/5·25-s − 0.392·26-s + 0.962·27-s + 0.176·32-s − 0.696·33-s − 1/3·36-s − 0.657·37-s + 0.320·39-s + 0.603·44-s − 0.294·46-s + 2.33·47-s − 0.144·48-s − 5/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.937481869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.937481869\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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
−9.539413107084750280234399337698, −9.115034355755048929672905596631, −8.408486909011940238981299566497, −8.248270166406525333712391337291, −7.25038064772684706053554512263, −6.87482507907449229860442380234, −6.55862357806514374286079019861, −5.88794573838587439349205533248, −5.28069074518443602737935578596, −5.09266829498719992624263414553, −4.05875927362791341229655398896, −3.86057015395193111701613139713, −2.85985483208439285249907021396, −2.24529612425880911912836760045, −0.985714903183510332642793788099,
0.985714903183510332642793788099, 2.24529612425880911912836760045, 2.85985483208439285249907021396, 3.86057015395193111701613139713, 4.05875927362791341229655398896, 5.09266829498719992624263414553, 5.28069074518443602737935578596, 5.88794573838587439349205533248, 6.55862357806514374286079019861, 6.87482507907449229860442380234, 7.25038064772684706053554512263, 8.248270166406525333712391337291, 8.408486909011940238981299566497, 9.115034355755048929672905596631, 9.539413107084750280234399337698