| L(s) = 1 | + 3-s + 3.35·7-s + 9-s − 0.740·11-s + 3.81·13-s + 1.61·17-s + 0.839·19-s + 3.35·21-s + 2.74·23-s + 27-s − 3.05·29-s + 1.77·31-s − 0.740·33-s − 4.89·37-s + 3.81·39-s + 5.11·41-s + 0.145·43-s + 2.65·47-s + 4.28·49-s + 1.61·51-s − 10.6·53-s + 0.839·57-s − 14.9·59-s + 14.9·61-s + 3.35·63-s + 8.41·67-s + 2.74·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.26·7-s + 0.333·9-s − 0.223·11-s + 1.05·13-s + 0.392·17-s + 0.192·19-s + 0.732·21-s + 0.571·23-s + 0.192·27-s − 0.566·29-s + 0.319·31-s − 0.128·33-s − 0.804·37-s + 0.611·39-s + 0.798·41-s + 0.0222·43-s + 0.387·47-s + 0.611·49-s + 0.226·51-s − 1.46·53-s + 0.111·57-s − 1.94·59-s + 1.91·61-s + 0.423·63-s + 1.02·67-s + 0.329·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.941149922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941149922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 0.740T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 0.839T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 0.145T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 8.41T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 79 | \( 1 - 0.0327T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 + 3.11T + 97T^{2} \) |
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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
−8.619081700531555395037285181248, −8.032495423495040099997824397076, −7.47311039673412969479342789029, −6.52364970784281231616938771922, −5.55824221003085886156814727246, −4.84633127318268902696500725362, −3.96258262190347648521034249658, −3.11362963645510737408212883653, −1.99330523325770970324148244167, −1.12428973580978937860598813942,
1.12428973580978937860598813942, 1.99330523325770970324148244167, 3.11362963645510737408212883653, 3.96258262190347648521034249658, 4.84633127318268902696500725362, 5.55824221003085886156814727246, 6.52364970784281231616938771922, 7.47311039673412969479342789029, 8.032495423495040099997824397076, 8.619081700531555395037285181248
