L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 3·8-s + 3·9-s + 7·11-s + 2·12-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s + 12·19-s − 2·21-s + 7·22-s + 3·23-s + 6·24-s + 6·26-s + 4·27-s − 28-s − 15·29-s + 6·31-s − 32-s + 14·33-s − 2·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 1.06·8-s + 9-s + 2.11·11-s + 0.577·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 2.75·19-s − 0.436·21-s + 1.49·22-s + 0.625·23-s + 1.22·24-s + 1.17·26-s + 0.769·27-s − 0.188·28-s − 2.78·29-s + 1.07·31-s − 0.176·32-s + 2.43·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.31938477\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.31938477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 15 T + 110 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 210 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.619839010287527726123048282061, −8.532021386644083971162390704030, −8.125885562189757696159443134353, −7.32494797928465343466788187974, −7.21337777476355516420931328990, −7.10577924616050045406455547789, −6.65399523212878187514296852989, −6.12231734038658971106213126440, −5.72564681948181904840733328703, −5.45581911451406974326889384389, −4.69152801212127144907335384724, −4.54844311995279887696006024032, −3.95765226264758390153285765843, −3.46829556796645367684705152729, −3.37489841745021752124814137040, −3.24310939014855361679617182597, −2.22820754835305245913204896375, −1.58207226202124541382049686365, −1.56826269906311933645701924599, −0.882930258623306524822922614315,
0.882930258623306524822922614315, 1.56826269906311933645701924599, 1.58207226202124541382049686365, 2.22820754835305245913204896375, 3.24310939014855361679617182597, 3.37489841745021752124814137040, 3.46829556796645367684705152729, 3.95765226264758390153285765843, 4.54844311995279887696006024032, 4.69152801212127144907335384724, 5.45581911451406974326889384389, 5.72564681948181904840733328703, 6.12231734038658971106213126440, 6.65399523212878187514296852989, 7.10577924616050045406455547789, 7.21337777476355516420931328990, 7.32494797928465343466788187974, 8.125885562189757696159443134353, 8.532021386644083971162390704030, 8.619839010287527726123048282061