| L(s) = 1 | − 2·2-s − 15·5-s + 35·7-s + 8·8-s + 30·10-s − 9·11-s − 176·13-s − 70·14-s − 16·16-s − 84·17-s − 104·19-s + 18·22-s − 84·23-s + 125·25-s + 352·26-s − 102·29-s − 185·31-s + 168·34-s − 525·35-s − 44·37-s + 208·38-s − 120·40-s + 336·41-s + 652·43-s + 168·46-s − 138·47-s + 882·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 1.88·7-s + 0.353·8-s + 0.948·10-s − 0.246·11-s − 3.75·13-s − 1.33·14-s − 1/4·16-s − 1.19·17-s − 1.25·19-s + 0.174·22-s − 0.761·23-s + 25-s + 2.65·26-s − 0.653·29-s − 1.07·31-s + 0.847·34-s − 2.53·35-s − 0.195·37-s + 0.887·38-s − 0.474·40-s + 1.27·41-s + 2.31·43-s + 0.538·46-s − 0.428·47-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2839192599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2839192599\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 p T + 4 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T - 1250 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 104 T + 3957 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 51 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 185 T + 4434 T^{2} + 185 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 44 T - 48717 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 326 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 138 T - 84779 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 639 T + 259444 T^{2} - 639 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 159 T - 180098 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 722 T + 294303 T^{2} + 722 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 166 T - 273207 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1086 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 218 T - 341493 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 583 T - 153150 T^{2} - 583 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 597 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1038 T + 372475 T^{2} + 1038 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 169 T + p^{3} T^{2} )^{2} \) |
| 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
−13.85900650157830376361676143775, −12.38308850917137797759883821071, −12.19361391273136859432421604188, −11.35628776488920794173128048962, −11.32233358802394741211733964735, −10.36045697894851948798979572405, −10.34990424577050086924240803354, −9.156943194472649256854722982684, −9.051209869662086813462304989477, −8.237245435564005651070826518422, −7.67162902166206066772938522486, −7.34799184073233104964691281445, −7.23451551568955188693241924832, −5.77811076928878933047127305554, −4.95622830621082799261887348208, −4.39188351868946152677664610665, −4.27809593741076060272821408445, −2.45856932309852144474643919835, −2.08132417478727407476292770793, −0.29663420005227452098801960331,
0.29663420005227452098801960331, 2.08132417478727407476292770793, 2.45856932309852144474643919835, 4.27809593741076060272821408445, 4.39188351868946152677664610665, 4.95622830621082799261887348208, 5.77811076928878933047127305554, 7.23451551568955188693241924832, 7.34799184073233104964691281445, 7.67162902166206066772938522486, 8.237245435564005651070826518422, 9.051209869662086813462304989477, 9.156943194472649256854722982684, 10.34990424577050086924240803354, 10.36045697894851948798979572405, 11.32233358802394741211733964735, 11.35628776488920794173128048962, 12.19361391273136859432421604188, 12.38308850917137797759883821071, 13.85900650157830376361676143775
