L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 25-s − 32-s + 36-s − 41-s − 2·43-s − 49-s − 50-s − 2·59-s + 64-s − 72-s − 2·73-s + 81-s + 82-s + 2·83-s + 2·86-s + 98-s + 100-s + 2·107-s − 2·113-s + 2·118-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 25-s − 32-s + 36-s − 41-s − 2·43-s − 49-s − 50-s − 2·59-s + 64-s − 72-s − 2·73-s + 81-s + 82-s + 2·83-s + 2·86-s + 98-s + 100-s + 2·107-s − 2·113-s + 2·118-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5632208795\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5632208795\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 41 | \( 1 + T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.66752597725911961624971455992, −10.64392297554471630400677505911, −9.980175470070455659993855969363, −9.069570123740443724390404064793, −8.105031387169368766400681276590, −7.14312272258317708388536898329, −6.34687988785922521290230471969, −4.85894487317697649364386311407, −3.25779803806112285509082430856, −1.63266017683856568882926995905,
1.63266017683856568882926995905, 3.25779803806112285509082430856, 4.85894487317697649364386311407, 6.34687988785922521290230471969, 7.14312272258317708388536898329, 8.105031387169368766400681276590, 9.069570123740443724390404064793, 9.980175470070455659993855969363, 10.64392297554471630400677505911, 11.66752597725911961624971455992