| L(s) = 1 | + 3-s + 3.51·5-s + 7-s + 9-s + 1.74·11-s + 6.33·13-s + 3.51·15-s − 5.94·17-s − 1.74·19-s + 21-s − 23-s + 7.33·25-s + 27-s − 5.68·29-s − 7.94·31-s + 1.74·33-s + 3.51·35-s + 1.53·37-s + 6.33·39-s + 12.1·41-s − 6.43·43-s + 3.51·45-s − 3.59·47-s + 49-s − 5.94·51-s + 12.9·53-s + 6.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.57·5-s + 0.377·7-s + 0.333·9-s + 0.525·11-s + 1.75·13-s + 0.906·15-s − 1.44·17-s − 0.399·19-s + 0.218·21-s − 0.208·23-s + 1.46·25-s + 0.192·27-s − 1.05·29-s − 1.42·31-s + 0.303·33-s + 0.593·35-s + 0.252·37-s + 1.01·39-s + 1.89·41-s − 0.981·43-s + 0.523·45-s − 0.524·47-s + 0.142·49-s − 0.832·51-s + 1.77·53-s + 0.825·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.216060762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.216060762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 8.69T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 4.46T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 - 7.42T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 - 4.18T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−8.082243838947892189623514803689, −6.94780137012735019449757008989, −6.51349206185616921169893333106, −5.79967427452803150834024181265, −5.23579777329805449339148490397, −4.09038538584592480744241090129, −3.66730676847375082168734886218, −2.31790573184105091463959102782, −1.98849833132928814604858362612, −1.06510887790488622720664097977,
1.06510887790488622720664097977, 1.98849833132928814604858362612, 2.31790573184105091463959102782, 3.66730676847375082168734886218, 4.09038538584592480744241090129, 5.23579777329805449339148490397, 5.79967427452803150834024181265, 6.51349206185616921169893333106, 6.94780137012735019449757008989, 8.082243838947892189623514803689
