| L(s) = 1 | + 25.1·3-s + 25·5-s − 49·7-s + 390.·9-s + 260.·11-s + 96.1·13-s + 629.·15-s − 328.·17-s + 560.·19-s − 1.23e3·21-s + 2.85e3·23-s + 625·25-s + 3.70e3·27-s + 4.17e3·29-s + 573.·31-s + 6.55e3·33-s − 1.22e3·35-s + 1.36e4·37-s + 2.42e3·39-s − 4.17e3·41-s + 5.02e3·43-s + 9.75e3·45-s − 1.20e4·47-s + 2.40e3·49-s − 8.26e3·51-s + 1.65e4·53-s + 6.51e3·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.447·5-s − 0.377·7-s + 1.60·9-s + 0.649·11-s + 0.157·13-s + 0.721·15-s − 0.275·17-s + 0.356·19-s − 0.610·21-s + 1.12·23-s + 0.200·25-s + 0.978·27-s + 0.920·29-s + 0.107·31-s + 1.04·33-s − 0.169·35-s + 1.63·37-s + 0.254·39-s − 0.387·41-s + 0.414·43-s + 0.718·45-s − 0.794·47-s + 0.142·49-s − 0.445·51-s + 0.811·53-s + 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.288798830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288798830\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 25.1T + 243T^{2} \) |
| 11 | \( 1 - 260.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 96.1T + 3.71e5T^{2} \) |
| 17 | \( 1 + 328.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 560.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 573.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.26e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81626938250773577290572222031, −9.661859022266648393372242455941, −9.164841206354382788615851662756, −8.295691267544376732587454430492, −7.25056847154449097280560911387, −6.23163925891571203453909847899, −4.59987628050868114513223644318, −3.39440550236445549284780588541, −2.52569653463138567642743720399, −1.21989225431456724424030529644,
1.21989225431456724424030529644, 2.52569653463138567642743720399, 3.39440550236445549284780588541, 4.59987628050868114513223644318, 6.23163925891571203453909847899, 7.25056847154449097280560911387, 8.295691267544376732587454430492, 9.164841206354382788615851662756, 9.661859022266648393372242455941, 10.81626938250773577290572222031
