| L(s) = 1 | − 4·2-s + 16·4-s + 142·7-s − 64·8-s − 777·11-s − 884·13-s − 568·14-s + 256·16-s − 27·17-s + 1.14e3·19-s + 3.10e3·22-s + 1.85e3·23-s + 3.53e3·26-s + 2.27e3·28-s + 4.92e3·29-s + 1.80e3·31-s − 1.02e3·32-s + 108·34-s − 1.31e4·37-s − 4.58e3·38-s + 1.51e4·41-s − 7.84e3·43-s − 1.24e4·44-s − 7.41e3·46-s − 6.73e3·47-s + 3.35e3·49-s − 1.41e4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.09·7-s − 0.353·8-s − 1.93·11-s − 1.45·13-s − 0.774·14-s + 1/4·16-s − 0.0226·17-s + 0.727·19-s + 1.36·22-s + 0.730·23-s + 1.02·26-s + 0.547·28-s + 1.08·29-s + 0.336·31-s − 0.176·32-s + 0.0160·34-s − 1.58·37-s − 0.514·38-s + 1.40·41-s − 0.646·43-s − 0.968·44-s − 0.516·46-s − 0.444·47-s + 0.199·49-s − 0.725·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.218565049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218565049\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 142 T + p^{5} T^{2} \) |
| 11 | \( 1 + 777 T + p^{5} T^{2} \) |
| 13 | \( 1 + 68 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 27 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1145 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1854 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4920 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1802 T + p^{5} T^{2} \) |
| 37 | \( 1 + 13178 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15123 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7844 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6732 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3414 T + p^{5} T^{2} \) |
| 59 | \( 1 + 33960 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47402 T + p^{5} T^{2} \) |
| 67 | \( 1 - 13177 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 - 59821 T + p^{5} T^{2} \) |
| 79 | \( 1 - 75830 T + p^{5} T^{2} \) |
| 83 | \( 1 - 46299 T + p^{5} T^{2} \) |
| 89 | \( 1 - 30585 T + p^{5} T^{2} \) |
| 97 | \( 1 + 104018 T + p^{5} T^{2} \) |
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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
−10.32385928379703149973303403873, −9.461471525420008007664606178655, −8.244136771050783048640768385565, −7.77936313962986678871960233001, −6.92049571328764833213607850506, −5.30126891918805094451814331580, −4.86805368169635886181913096340, −2.94087873912969433873138887537, −2.06600820055149981296405460772, −0.61518788594255754770407714526,
0.61518788594255754770407714526, 2.06600820055149981296405460772, 2.94087873912969433873138887537, 4.86805368169635886181913096340, 5.30126891918805094451814331580, 6.92049571328764833213607850506, 7.77936313962986678871960233001, 8.244136771050783048640768385565, 9.461471525420008007664606178655, 10.32385928379703149973303403873
