| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 8-s + 3·9-s − 10-s + 6·11-s + 8·13-s − 15-s − 16-s + 3·18-s + 2·19-s + 6·22-s + 3·23-s − 24-s + 8·26-s + 8·27-s − 6·29-s − 30-s + 8·31-s + 6·33-s + 4·37-s + 2·38-s + 8·39-s + 40-s − 18·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 9-s − 0.316·10-s + 1.80·11-s + 2.21·13-s − 0.258·15-s − 1/4·16-s + 0.707·18-s + 0.458·19-s + 1.27·22-s + 0.625·23-s − 0.204·24-s + 1.56·26-s + 1.53·27-s − 1.11·29-s − 0.182·30-s + 1.43·31-s + 1.04·33-s + 0.657·37-s + 0.324·38-s + 1.28·39-s + 0.158·40-s − 2.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.671397104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.671397104\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.31180263597056331023550613017, −10.89567706423197120681701405633, −10.09307224558222568342317572287, −9.988001217948260734409184863940, −9.368678656154217523830540462538, −8.658619029853119201835993274611, −8.565910687306719924243848800689, −8.387264978175214957996127455161, −7.33729532542365812324902201293, −7.07454466644121467035327596248, −6.40509685473105616021603003070, −6.31936969058313208499300326232, −5.52512359678824907726235184739, −4.84651664594467246665156180400, −4.18753996635165214267693401800, −4.02849380592388524570548907508, −3.22982256778607155970356288366, −3.15367730429890774651553572651, −1.55250784867597881060993843962, −1.30864786711073282769698328730,
1.30864786711073282769698328730, 1.55250784867597881060993843962, 3.15367730429890774651553572651, 3.22982256778607155970356288366, 4.02849380592388524570548907508, 4.18753996635165214267693401800, 4.84651664594467246665156180400, 5.52512359678824907726235184739, 6.31936969058313208499300326232, 6.40509685473105616021603003070, 7.07454466644121467035327596248, 7.33729532542365812324902201293, 8.387264978175214957996127455161, 8.565910687306719924243848800689, 8.658619029853119201835993274611, 9.368678656154217523830540462538, 9.988001217948260734409184863940, 10.09307224558222568342317572287, 10.89567706423197120681701405633, 11.31180263597056331023550613017
