L(s) = 1 | + 2·3-s − 7-s + 9-s + 12·17-s − 2·21-s − 10·25-s − 4·27-s + 4·37-s + 12·41-s − 16·43-s + 24·47-s + 49-s + 24·51-s + 12·59-s − 63-s + 8·67-s − 20·75-s − 16·79-s − 11·81-s + 12·83-s − 12·89-s + 4·109-s + 8·111-s − 12·119-s − 22·121-s + 24·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 2.91·17-s − 0.436·21-s − 2·25-s − 0.769·27-s + 0.657·37-s + 1.87·41-s − 2.43·43-s + 3.50·47-s + 1/7·49-s + 3.36·51-s + 1.56·59-s − 0.125·63-s + 0.977·67-s − 2.30·75-s − 1.80·79-s − 1.22·81-s + 1.31·83-s − 1.27·89-s + 0.383·109-s + 0.759·111-s − 1.10·119-s − 2·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.067143272\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.067143272\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−8.182643116839158569055135937831, −7.88115201382147028507134927026, −7.39293674981761532961441986375, −7.29668061464756137953129595214, −6.43902889849944115155691598253, −5.83259812621372959122100845675, −5.62168100149868099368912779374, −5.25590580602395553760699468065, −4.24465289176609138955904372614, −3.82418745638366191622617161116, −3.56550924639142318669266713404, −2.84501304641458537647904209821, −2.48514181668809031419660323366, −1.67921161671410140538744564928, −0.818365395076437722978448101852,
0.818365395076437722978448101852, 1.67921161671410140538744564928, 2.48514181668809031419660323366, 2.84501304641458537647904209821, 3.56550924639142318669266713404, 3.82418745638366191622617161116, 4.24465289176609138955904372614, 5.25590580602395553760699468065, 5.62168100149868099368912779374, 5.83259812621372959122100845675, 6.43902889849944115155691598253, 7.29668061464756137953129595214, 7.39293674981761532961441986375, 7.88115201382147028507134927026, 8.182643116839158569055135937831