L(s) = 1 | + 5·7-s + 2·11-s + 2·13-s − 2·17-s + 5·19-s − 6·23-s + 5·25-s + 16·29-s − 3·31-s + 9·37-s − 4·41-s − 2·43-s − 8·47-s + 18·49-s + 6·53-s − 6·59-s + 2·61-s − 5·67-s + 8·71-s + 11·73-s + 10·77-s − 5·79-s + 12·89-s + 10·91-s + 36·97-s − 6·101-s − 11·103-s + ⋯ |
L(s) = 1 | + 1.88·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 1.14·19-s − 1.25·23-s + 25-s + 2.97·29-s − 0.538·31-s + 1.47·37-s − 0.624·41-s − 0.304·43-s − 1.16·47-s + 18/7·49-s + 0.824·53-s − 0.781·59-s + 0.256·61-s − 0.610·67-s + 0.949·71-s + 1.28·73-s + 1.13·77-s − 0.562·79-s + 1.27·89-s + 1.04·91-s + 3.65·97-s − 0.597·101-s − 1.08·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.336744018\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.336744018\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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
−9.208110597932124863375837082776, −8.896128638598400707479821494544, −8.380300615545283683005214063971, −8.313008807830177723787394250935, −7.88331914244165383674418672591, −7.50528123297249961836126995042, −6.98529144610785914249417158080, −6.63327296516652306383051610150, −6.09664405674657695851198621875, −5.90633581703496709628859743985, −5.07129258196558301496825491902, −4.97941412507888957099089621112, −4.37059897616797111250226751603, −4.35423131221347129320641975682, −3.30444471021817680671187492401, −3.28961494686958588471894368565, −2.18051935450950008292518745423, −2.09987401460661292316185846399, −1.05182543295580171154941226554, −0.997018932130729520798242311801,
0.997018932130729520798242311801, 1.05182543295580171154941226554, 2.09987401460661292316185846399, 2.18051935450950008292518745423, 3.28961494686958588471894368565, 3.30444471021817680671187492401, 4.35423131221347129320641975682, 4.37059897616797111250226751603, 4.97941412507888957099089621112, 5.07129258196558301496825491902, 5.90633581703496709628859743985, 6.09664405674657695851198621875, 6.63327296516652306383051610150, 6.98529144610785914249417158080, 7.50528123297249961836126995042, 7.88331914244165383674418672591, 8.313008807830177723787394250935, 8.380300615545283683005214063971, 8.896128638598400707479821494544, 9.208110597932124863375837082776