| L(s) = 1 | + 3-s − 7-s − 13-s + 2·19-s − 21-s + 2·25-s − 27-s + 2·31-s − 2·37-s − 39-s − 43-s + 49-s + 2·57-s + 61-s − 67-s − 2·73-s + 2·75-s + 2·79-s − 81-s + 91-s + 2·93-s + 97-s + 2·103-s − 2·109-s − 2·111-s − 121-s + 127-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s − 13-s + 2·19-s − 21-s + 2·25-s − 27-s + 2·31-s − 2·37-s − 39-s − 43-s + 49-s + 2·57-s + 61-s − 67-s − 2·73-s + 2·75-s + 2·79-s − 81-s + 91-s + 2·93-s + 97-s + 2·103-s − 2·109-s − 2·111-s − 121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.000704343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000704343\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−10.97129559742215482040433422081, −10.32018202553473976428829763955, −9.981085007977267980040519429716, −9.931116091747346110423443473801, −9.161734658914559992170500016083, −8.819517375467821617261056172790, −8.706343166259501100413986556692, −7.905440660939254498211302819696, −7.52829590889521030325909695006, −7.18696686612219393493168256722, −6.54061499592056824897355517689, −6.35739422124495901714742421690, −5.28998269265951728654071309050, −5.24019114261790407986474765330, −4.58052234750667136602277376957, −3.71554517260195559653698089902, −3.23304083245014877016300954431, −2.88839019527895316305330121017, −2.39082741539602521707451450714, −1.23506609194945257266445349374,
1.23506609194945257266445349374, 2.39082741539602521707451450714, 2.88839019527895316305330121017, 3.23304083245014877016300954431, 3.71554517260195559653698089902, 4.58052234750667136602277376957, 5.24019114261790407986474765330, 5.28998269265951728654071309050, 6.35739422124495901714742421690, 6.54061499592056824897355517689, 7.18696686612219393493168256722, 7.52829590889521030325909695006, 7.905440660939254498211302819696, 8.706343166259501100413986556692, 8.819517375467821617261056172790, 9.161734658914559992170500016083, 9.931116091747346110423443473801, 9.981085007977267980040519429716, 10.32018202553473976428829763955, 10.97129559742215482040433422081
