| L(s) = 1 | + 5·5-s − 5.08·7-s − 58.3·11-s + 21.2·13-s − 68.8·17-s + 40.8·19-s + 144.·23-s + 25·25-s − 220.·29-s − 291.·31-s − 25.4·35-s + 260.·37-s − 169.·41-s + 438.·43-s + 255.·47-s − 317.·49-s + 214.·53-s − 291.·55-s − 331.·59-s + 54.9·61-s + 106.·65-s − 758.·67-s + 904.·71-s + 866.·73-s + 296.·77-s − 206.·79-s + 463.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.274·7-s − 1.59·11-s + 0.452·13-s − 0.982·17-s + 0.492·19-s + 1.30·23-s + 0.200·25-s − 1.40·29-s − 1.68·31-s − 0.122·35-s + 1.15·37-s − 0.646·41-s + 1.55·43-s + 0.792·47-s − 0.924·49-s + 0.556·53-s − 0.714·55-s − 0.731·59-s + 0.115·61-s + 0.202·65-s − 1.38·67-s + 1.51·71-s + 1.38·73-s + 0.438·77-s − 0.294·79-s + 0.612·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.680276644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680276644\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| good | 7 | \( 1 + 5.08T + 343T^{2} \) |
| 11 | \( 1 + 58.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 255.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 54.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 758.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 866.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 206.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 463.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 601.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 229.T + 9.12e5T^{2} \) |
| 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
−8.976322283948955632381476541839, −7.78334205277395964477329231318, −7.31258320542622850644462473984, −6.29301487779999469931204789154, −5.50989989442889908482387859375, −4.89600460652895144887004125287, −3.72109248109049723432458723942, −2.77308847252182522952751599157, −1.94534482769716810466713103830, −0.56695036829938167538167713089,
0.56695036829938167538167713089, 1.94534482769716810466713103830, 2.77308847252182522952751599157, 3.72109248109049723432458723942, 4.89600460652895144887004125287, 5.50989989442889908482387859375, 6.29301487779999469931204789154, 7.31258320542622850644462473984, 7.78334205277395964477329231318, 8.976322283948955632381476541839
