L(s) = 1 | − 2i·13-s + 25-s − 2i·37-s + 49-s + 2i·61-s + 2·73-s − 2·97-s − 2i·109-s + ⋯ |
L(s) = 1 | − 2i·13-s + 25-s − 2i·37-s + 49-s + 2i·61-s + 2·73-s − 2·97-s − 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.157134780\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.157134780\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 2iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.030360332210759474978608301349, −8.306522801608302811399082525584, −7.59670940442531178970512800069, −6.88036566963366407484521607269, −5.74614482141289107057794243746, −5.35516703300717602532838583971, −4.22310695014203753438899659805, −3.25576330057765085835529047619, −2.42610052844155881412945048433, −0.871239430364633848829850410997,
1.43762214026917881073060627354, 2.49320609828736547236567061822, 3.64167956491301591203916541276, 4.51324427484928795713985504553, 5.20531561412361022727792090914, 6.53062603387413204559482472777, 6.69352018234265738754400787846, 7.77869817667419196365074396252, 8.618582661939724690539161761209, 9.256138654178166197316001342571