| L(s) = 1 | + 28·9-s + 136·17-s − 50·25-s + 1.08e3·41-s − 652·49-s + 4.31e3·73-s − 870·81-s − 3.56e3·89-s − 1.01e3·97-s + 6.79e3·113-s + 5.00e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3.80e3·153-s + 157-s + 163-s + 167-s + 5.90e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.03·9-s + 1.94·17-s − 2/5·25-s + 4.11·41-s − 1.90·49-s + 6.91·73-s − 1.19·81-s − 4.23·89-s − 1.06·97-s + 5.65·113-s + 3.75·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 2.01·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.68·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.277102519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.277102519\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 326 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2502 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2950 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 3478 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 17574 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 24122 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 57058 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58870 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 129986 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 190006 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 231190 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 404998 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 391462 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 64114 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 299662 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1078 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 908638 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 81426 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 p T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{4} \) |
| 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
−6.76334199834488717695190898256, −6.14033637246231461809883166794, −5.98123965967583634683919885294, −5.86877200099169365062526465349, −5.66908079039429252359100058802, −5.60911064677210856813523532429, −5.01686517577959828682513994859, −4.88824420254756755105151917381, −4.88785944602431937320950888295, −4.46125160074986236842289185107, −4.10801582112110541595441152344, −3.97324091652113089137391785503, −3.95979359141391801373763779250, −3.32734011147698070760946091363, −3.32378646258497278403151611240, −3.10948735290792275387142931998, −2.69418925867076843712065486338, −2.36328466462289636218519248887, −2.10375323762857487357295089433, −1.92204110157228652596645084763, −1.48762615716734369709373136345, −1.06890589330679983074728970496, −0.903926971672448734952476180696, −0.78024702920185381145338010395, −0.23383919445252401361224454736,
0.23383919445252401361224454736, 0.78024702920185381145338010395, 0.903926971672448734952476180696, 1.06890589330679983074728970496, 1.48762615716734369709373136345, 1.92204110157228652596645084763, 2.10375323762857487357295089433, 2.36328466462289636218519248887, 2.69418925867076843712065486338, 3.10948735290792275387142931998, 3.32378646258497278403151611240, 3.32734011147698070760946091363, 3.95979359141391801373763779250, 3.97324091652113089137391785503, 4.10801582112110541595441152344, 4.46125160074986236842289185107, 4.88785944602431937320950888295, 4.88824420254756755105151917381, 5.01686517577959828682513994859, 5.60911064677210856813523532429, 5.66908079039429252359100058802, 5.86877200099169365062526465349, 5.98123965967583634683919885294, 6.14033637246231461809883166794, 6.76334199834488717695190898256
