| L(s) = 1 | + 2·7-s + 13-s + 6·19-s + 2·23-s + 4·29-s − 8·31-s + 10·37-s + 2·41-s − 12·43-s + 4·47-s − 3·49-s + 6·53-s + 8·59-s − 2·61-s − 4·67-s − 4·83-s + 14·89-s + 2·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.277·13-s + 1.37·19-s + 0.417·23-s + 0.742·29-s − 1.43·31-s + 1.64·37-s + 0.312·41-s − 1.82·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.488·67-s − 0.439·83-s + 1.48·89-s + 0.209·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.287151880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.287151880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.80245182648860, −13.31413542164769, −12.98333736365555, −12.26715728568420, −11.78950254914310, −11.31909638079936, −11.07783944425715, −10.28098744243300, −9.944606449914146, −9.301507219510462, −8.816006845268740, −8.351039183356132, −7.642811564898447, −7.450600959289896, −6.748625689554489, −6.127997557142221, −5.557842408707391, −5.022101784679517, −4.614515663594701, −3.825338637855492, −3.325352212555623, −2.650409449018221, −1.923604032044104, −1.251416578875354, −0.6228589894023504,
0.6228589894023504, 1.251416578875354, 1.923604032044104, 2.650409449018221, 3.325352212555623, 3.825338637855492, 4.614515663594701, 5.022101784679517, 5.557842408707391, 6.127997557142221, 6.748625689554489, 7.450600959289896, 7.642811564898447, 8.351039183356132, 8.816006845268740, 9.301507219510462, 9.944606449914146, 10.28098744243300, 11.07783944425715, 11.31909638079936, 11.78950254914310, 12.26715728568420, 12.98333736365555, 13.31413542164769, 13.80245182648860
