L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 5-s + 4·6-s − 4·7-s + 4·8-s + 3·9-s + 2·10-s + 6·12-s + 2·13-s − 8·14-s + 2·15-s + 5·16-s + 6·18-s + 19-s + 3·20-s − 8·21-s − 6·23-s + 8·24-s + 4·26-s + 10·27-s − 12·28-s − 6·29-s + 4·30-s + 4·31-s + 6·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 1.51·7-s + 1.41·8-s + 9-s + 0.632·10-s + 1.73·12-s + 0.554·13-s − 2.13·14-s + 0.516·15-s + 5/4·16-s + 1.41·18-s + 0.229·19-s + 0.670·20-s − 1.74·21-s − 1.25·23-s + 1.63·24-s + 0.784·26-s + 1.92·27-s − 2.26·28-s − 1.11·29-s + 0.730·30-s + 0.718·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.833341764\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.833341764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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
−10.33287846109460160744479203469, −10.01460145085722174837698440469, −9.246504691903543852532236764118, −9.169900392644678030316739246238, −8.876782493895710555522447460816, −7.902923246489733380569872508626, −7.75775083318443427761830809067, −7.36703791789534490790135418552, −6.63296399265105306770603524894, −6.39383527805987100205186036517, −5.99255138310782564616561657394, −5.66614868031204031614483236965, −4.86886340838849018606924092821, −4.39683051481409444885736219445, −3.90317074247142605885648743447, −3.35221635680914057820678022251, −3.19217752593871883984796418345, −2.35625348708073913543745842680, −2.15220438825430103838813756675, −1.03112870002132224359559612717,
1.03112870002132224359559612717, 2.15220438825430103838813756675, 2.35625348708073913543745842680, 3.19217752593871883984796418345, 3.35221635680914057820678022251, 3.90317074247142605885648743447, 4.39683051481409444885736219445, 4.86886340838849018606924092821, 5.66614868031204031614483236965, 5.99255138310782564616561657394, 6.39383527805987100205186036517, 6.63296399265105306770603524894, 7.36703791789534490790135418552, 7.75775083318443427761830809067, 7.902923246489733380569872508626, 8.876782493895710555522447460816, 9.169900392644678030316739246238, 9.246504691903543852532236764118, 10.01460145085722174837698440469, 10.33287846109460160744479203469