| L(s) = 1 | − 2.35·2-s + 3.53·4-s − 3.60·5-s + 0.184·7-s − 3.60·8-s + 8.47·10-s + 3.16·11-s + 6.06·13-s − 0.434·14-s + 1.41·16-s + 3.31·17-s − 12.7·20-s − 7.45·22-s − 4.42·23-s + 7.98·25-s − 14.2·26-s + 0.652·28-s + 7.58·29-s + 7.41·31-s + 3.88·32-s − 7.80·34-s − 0.665·35-s + 1.77·37-s + 12.9·40-s − 2.21·41-s + 6.75·43-s + 11.1·44-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.76·4-s − 1.61·5-s + 0.0698·7-s − 1.27·8-s + 2.68·10-s + 0.955·11-s + 1.68·13-s − 0.116·14-s + 0.352·16-s + 0.805·17-s − 2.84·20-s − 1.58·22-s − 0.921·23-s + 1.59·25-s − 2.79·26-s + 0.123·28-s + 1.40·29-s + 1.33·31-s + 0.687·32-s − 1.33·34-s − 0.112·35-s + 0.291·37-s + 2.05·40-s − 0.346·41-s + 1.03·43-s + 1.68·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6893579646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6893579646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.184T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 0.150T + 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 + 5.92T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 - 7.30T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 - 0.837T + 97T^{2} \) |
| 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
−8.482067043854657774037695332046, −8.128880472882712110956858321677, −7.55193859995257070380371596853, −6.61885261418033716886467982372, −6.14223668739252574295931972557, −4.55989231069890189171321414529, −3.82707746950328978179964741885, −2.98133672115075024837743682360, −1.43639638800924185331670376689, −0.71670587943041681662619772859,
0.71670587943041681662619772859, 1.43639638800924185331670376689, 2.98133672115075024837743682360, 3.82707746950328978179964741885, 4.55989231069890189171321414529, 6.14223668739252574295931972557, 6.61885261418033716886467982372, 7.55193859995257070380371596853, 8.128880472882712110956858321677, 8.482067043854657774037695332046
