L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s + 16-s + 18-s − 2·22-s + 24-s − 25-s + 27-s + 32-s − 2·33-s + 36-s − 2·44-s − 2·47-s + 48-s + 49-s − 50-s + 54-s − 2·61-s + 64-s − 2·66-s + 72-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s + 16-s + 18-s − 2·22-s + 24-s − 25-s + 27-s + 32-s − 2·33-s + 36-s − 2·44-s − 2·47-s + 48-s + 49-s − 50-s + 54-s − 2·61-s + 64-s − 2·66-s + 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.828944251\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.828944251\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−9.409026819030253402152017068599, −8.210781164323262641046444724904, −7.81344714850669165146848639095, −7.10326380417237278822140142446, −6.05523110315333649129783722238, −5.17015322538927603798048442380, −4.44788671815030718172705710070, −3.40351519948223835697325871832, −2.70002137765165628169346249819, −1.85390657607385376562342581241,
1.85390657607385376562342581241, 2.70002137765165628169346249819, 3.40351519948223835697325871832, 4.44788671815030718172705710070, 5.17015322538927603798048442380, 6.05523110315333649129783722238, 7.10326380417237278822140142446, 7.81344714850669165146848639095, 8.210781164323262641046444724904, 9.409026819030253402152017068599