L(s) = 1 | − 4-s + 2·7-s + 16-s − 2·19-s − 25-s − 2·28-s − 31-s + 3·49-s − 64-s − 2·67-s + 2·76-s + 2·97-s + 100-s − 2·103-s + 2·109-s + 2·112-s + ⋯ |
L(s) = 1 | − 4-s + 2·7-s + 16-s − 2·19-s − 25-s − 2·28-s − 31-s + 3·49-s − 64-s − 2·67-s + 2·76-s + 2·97-s + 100-s − 2·103-s + 2·109-s + 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7216711995\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7216711995\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 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
−12.08018341524497221045736657749, −11.09521140809559409479802410105, −10.33599727177141278791702208469, −8.992590991605716742617066459935, −8.348448212594379224261787842844, −7.54688362034994282287623546266, −5.85143602856598050045515693047, −4.79360867515447630427111469638, −4.06366238061491538956308440142, −1.86486858667779050269360380283,
1.86486858667779050269360380283, 4.06366238061491538956308440142, 4.79360867515447630427111469638, 5.85143602856598050045515693047, 7.54688362034994282287623546266, 8.348448212594379224261787842844, 8.992590991605716742617066459935, 10.33599727177141278791702208469, 11.09521140809559409479802410105, 12.08018341524497221045736657749