| L(s) = 1 | + 5·5-s + 13.6·7-s + 27.8·11-s + 59.2·13-s + 87.5·17-s + 161.·19-s − 131.·23-s + 25·25-s + 151.·29-s − 228.·31-s + 68.0·35-s + 236.·37-s + 226.·41-s − 152.·43-s + 129.·47-s − 157.·49-s − 492.·53-s + 139.·55-s − 879.·59-s − 434.·61-s + 296.·65-s + 539.·67-s + 65.4·71-s + 313.·73-s + 378.·77-s + 885.·79-s + 744.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.735·7-s + 0.762·11-s + 1.26·13-s + 1.24·17-s + 1.95·19-s − 1.19·23-s + 0.200·25-s + 0.969·29-s − 1.32·31-s + 0.328·35-s + 1.05·37-s + 0.863·41-s − 0.539·43-s + 0.402·47-s − 0.459·49-s − 1.27·53-s + 0.341·55-s − 1.93·59-s − 0.910·61-s + 0.565·65-s + 0.984·67-s + 0.109·71-s + 0.503·73-s + 0.560·77-s + 1.26·79-s + 0.985·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.491840494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.491840494\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 13.6T + 343T^{2} \) |
| 11 | \( 1 - 27.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 152.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 879.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 65.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 313.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 885.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 744.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 401.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
| 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
−9.225788947305782137305020865310, −8.044448286138618905669620592483, −7.69405255405834929280327170194, −6.45374924345850080555943734030, −5.80101678479406652965073078917, −5.00952050492938809641016792863, −3.89414165474150398189926870179, −3.09672076182492500502306314802, −1.63258190607942173383408126072, −1.02766742987637670277873935910,
1.02766742987637670277873935910, 1.63258190607942173383408126072, 3.09672076182492500502306314802, 3.89414165474150398189926870179, 5.00952050492938809641016792863, 5.80101678479406652965073078917, 6.45374924345850080555943734030, 7.69405255405834929280327170194, 8.044448286138618905669620592483, 9.225788947305782137305020865310
