| L(s) = 1 | + 2·2-s + 2.37·3-s + 4·4-s + 5·5-s + 4.74·6-s + 8·8-s − 21.3·9-s + 10·10-s + 49.8·11-s + 9.49·12-s + 42.1·13-s + 11.8·15-s + 16·16-s + 22.4·17-s − 42.7·18-s − 106.·19-s + 20·20-s + 99.7·22-s + 189.·23-s + 18.9·24-s + 25·25-s + 84.2·26-s − 114.·27-s − 52.5·29-s + 23.7·30-s + 154.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.456·3-s + 0.5·4-s + 0.447·5-s + 0.323·6-s + 0.353·8-s − 0.791·9-s + 0.316·10-s + 1.36·11-s + 0.228·12-s + 0.898·13-s + 0.204·15-s + 0.250·16-s + 0.320·17-s − 0.559·18-s − 1.28·19-s + 0.223·20-s + 0.966·22-s + 1.71·23-s + 0.161·24-s + 0.200·25-s + 0.635·26-s − 0.818·27-s − 0.336·29-s + 0.144·30-s + 0.893·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.169991170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.169991170\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.37T + 27T^{2} \) |
| 11 | \( 1 - 49.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 317.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 427.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 498.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 524.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 310.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 403.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 222.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 34.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 679.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 69.1T + 9.12e5T^{2} \) |
| 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
−10.89236556363556093797364182036, −9.456077711919471522642118287858, −8.862452947567715937710127264504, −7.85474332730230074287176002463, −6.47883830784183644529879194094, −6.06472773105630203308510659242, −4.71746855599811298293271357910, −3.65119818858329656325900816173, −2.64229085780678036981745288764, −1.26468853174248023551469801635,
1.26468853174248023551469801635, 2.64229085780678036981745288764, 3.65119818858329656325900816173, 4.71746855599811298293271357910, 6.06472773105630203308510659242, 6.47883830784183644529879194094, 7.85474332730230074287176002463, 8.862452947567715937710127264504, 9.456077711919471522642118287858, 10.89236556363556093797364182036
