L(s) = 1 | + 4-s + 2·7-s − 5·13-s + 16-s + 5·19-s + 25-s + 2·28-s + 6·31-s − 37-s + 2·43-s − 3·49-s − 5·52-s + 25·61-s + 64-s − 14·67-s + 15·73-s + 5·76-s + 5·79-s − 10·91-s − 10·97-s + 100-s − 15·103-s − 5·109-s + 2·112-s + 3·121-s + 6·124-s + 127-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.755·7-s − 1.38·13-s + 1/4·16-s + 1.14·19-s + 1/5·25-s + 0.377·28-s + 1.07·31-s − 0.164·37-s + 0.304·43-s − 3/7·49-s − 0.693·52-s + 3.20·61-s + 1/8·64-s − 1.71·67-s + 1.75·73-s + 0.573·76-s + 0.562·79-s − 1.04·91-s − 1.01·97-s + 1/10·100-s − 1.47·103-s − 0.478·109-s + 0.188·112-s + 3/11·121-s + 0.538·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 492156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 492156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.320902097\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.320902097\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 7 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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
−8.354009882390042347955773701647, −8.044178570859128834437004019585, −7.66513671790122837489604070631, −7.07045967322450886679876106851, −6.89843399027486271872061163621, −6.29622729719370616519634374934, −5.58000965969713619324716703521, −5.26628750181325296492793399603, −4.81531795628376667858261971664, −4.28354125360972107553657338118, −3.59817766153890748689952849091, −2.89260273168171248669020891087, −2.43792692987513023705092112252, −1.73777500476266676918806316722, −0.827320912311671096456745672017,
0.827320912311671096456745672017, 1.73777500476266676918806316722, 2.43792692987513023705092112252, 2.89260273168171248669020891087, 3.59817766153890748689952849091, 4.28354125360972107553657338118, 4.81531795628376667858261971664, 5.26628750181325296492793399603, 5.58000965969713619324716703521, 6.29622729719370616519634374934, 6.89843399027486271872061163621, 7.07045967322450886679876106851, 7.66513671790122837489604070631, 8.044178570859128834437004019585, 8.354009882390042347955773701647