L(s) = 1 | + 2-s + 4-s + 8-s + 16-s − 2·17-s + 32-s − 2·34-s + 49-s − 2·53-s − 2·61-s + 64-s − 2·68-s + 98-s − 2·106-s − 2·109-s + 2·113-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 8-s + 16-s − 2·17-s + 32-s − 2·34-s + 49-s − 2·53-s − 2·61-s + 64-s − 2·68-s + 98-s − 2·106-s − 2·109-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.784580502\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784580502\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 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
−10.68682024255314414976064779512, −9.527873296643808157033084528787, −8.580970433215067554859337494509, −7.55066885983855843486372146244, −6.69342016108698868656279910809, −5.99673558689662626326706314773, −4.85831092386239839105305591669, −4.18064729672705212396731265232, −2.98395527244364635313969528176, −1.89925069900852131232230684548,
1.89925069900852131232230684548, 2.98395527244364635313969528176, 4.18064729672705212396731265232, 4.85831092386239839105305591669, 5.99673558689662626326706314773, 6.69342016108698868656279910809, 7.55066885983855843486372146244, 8.580970433215067554859337494509, 9.527873296643808157033084528787, 10.68682024255314414976064779512