| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 4·13-s + 16-s − 2·17-s + 2·19-s + 2·23-s − 2·25-s + 6·27-s − 4·31-s − 3·36-s − 2·37-s + 8·39-s + 2·41-s − 2·43-s + 2·48-s − 49-s − 4·51-s − 4·52-s + 4·57-s + 2·59-s − 2·64-s + 2·68-s + 4·69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 4·13-s + 16-s − 2·17-s + 2·19-s + 2·23-s − 2·25-s + 6·27-s − 4·31-s − 3·36-s − 2·37-s + 8·39-s + 2·41-s − 2·43-s + 2·48-s − 49-s − 4·51-s − 4·52-s + 4·57-s + 2·59-s − 2·64-s + 2·68-s + 4·69-s + 2·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.157069873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.157069873\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$ | \( ( 1 + T )^{8} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.87278675372041595974301644151, −6.75040124717050228992262232807, −6.35395400830920339656943634574, −5.90286168760770248318018196157, −5.84650727057621951713300439644, −5.64115311147651199572814929322, −5.57543579479045702068846051341, −5.10952872245234090079413233872, −4.94760572378684714890200114083, −4.81284510964425801190112327088, −4.46081993106333233479484009585, −4.05807006694459404990572403087, −3.97654638410181062769900722985, −3.78192665371332889071938224063, −3.75365517112442613391113443739, −3.49073516583723840525631366810, −3.16579036118613786737294718931, −2.98368547026787120932660667826, −2.78094736662485428386819332448, −2.51203296321593883474084135356, −1.82662447873395131578870399265, −1.64454056673739804280620443546, −1.50933507309928468738952462602, −1.35697326076553220875024877427, −0.853013473635118497134698596767,
0.853013473635118497134698596767, 1.35697326076553220875024877427, 1.50933507309928468738952462602, 1.64454056673739804280620443546, 1.82662447873395131578870399265, 2.51203296321593883474084135356, 2.78094736662485428386819332448, 2.98368547026787120932660667826, 3.16579036118613786737294718931, 3.49073516583723840525631366810, 3.75365517112442613391113443739, 3.78192665371332889071938224063, 3.97654638410181062769900722985, 4.05807006694459404990572403087, 4.46081993106333233479484009585, 4.81284510964425801190112327088, 4.94760572378684714890200114083, 5.10952872245234090079413233872, 5.57543579479045702068846051341, 5.64115311147651199572814929322, 5.84650727057621951713300439644, 5.90286168760770248318018196157, 6.35395400830920339656943634574, 6.75040124717050228992262232807, 6.87278675372041595974301644151
