L(s) = 1 | − 4·5-s − 5·7-s + 6·11-s + 10·13-s + 2·17-s − 19-s − 6·23-s + 5·25-s + 3·31-s + 20·35-s − 3·37-s + 12·41-s + 10·43-s − 4·47-s + 18·49-s − 6·53-s − 24·55-s − 6·59-s + 2·61-s − 40·65-s − 7·67-s − 32·71-s + 3·73-s − 30·77-s − 11·79-s − 24·83-s − 8·85-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 1.88·7-s + 1.80·11-s + 2.77·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 25-s + 0.538·31-s + 3.38·35-s − 0.493·37-s + 1.87·41-s + 1.52·43-s − 0.583·47-s + 18/7·49-s − 0.824·53-s − 3.23·55-s − 0.781·59-s + 0.256·61-s − 4.96·65-s − 0.855·67-s − 3.79·71-s + 0.351·73-s − 3.41·77-s − 1.23·79-s − 2.63·83-s − 0.867·85-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
\(L(1)\) |
\(\approx\) |
\(1.172916437\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.172916437\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.705669078799358469279935259753, −8.874871154837142022026726549725, −8.748867424409151745968061019412, −8.228226317465367332758961560617, −7.76732161916545072025508817617, −7.32443263578210754350423006043, −7.12186113737288095602389529301, −6.44986139641714886459493366663, −6.14073094259087855165751058282, −5.91783845579342137278276328283, −5.81357087601946103823510898100, −4.40511545942400333726023419678, −4.29361359603048031755335932096, −3.93509239316057844557849880793, −3.68102080081483078686252494703, −3.09834762131747131662728026251, −2.98474324475393554791208926417, −1.71464561998985268449813331184, −1.15789215053590767730341613715, −0.45630287248820541482970382664,
0.45630287248820541482970382664, 1.15789215053590767730341613715, 1.71464561998985268449813331184, 2.98474324475393554791208926417, 3.09834762131747131662728026251, 3.68102080081483078686252494703, 3.93509239316057844557849880793, 4.29361359603048031755335932096, 4.40511545942400333726023419678, 5.81357087601946103823510898100, 5.91783845579342137278276328283, 6.14073094259087855165751058282, 6.44986139641714886459493366663, 7.12186113737288095602389529301, 7.32443263578210754350423006043, 7.76732161916545072025508817617, 8.228226317465367332758961560617, 8.748867424409151745968061019412, 8.874871154837142022026726549725, 9.705669078799358469279935259753