L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 3.05·11-s + 12-s + 1.64·13-s + 16-s − 2.90·17-s + 18-s + 2.21·19-s − 3.05·22-s + 1.78·23-s + 24-s + 1.64·26-s + 27-s + 5.58·29-s + 0.944·31-s + 32-s − 3.05·33-s − 2.90·34-s + 36-s − 10.7·37-s + 2.21·38-s + 1.64·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.921·11-s + 0.288·12-s + 0.455·13-s + 0.250·16-s − 0.705·17-s + 0.235·18-s + 0.507·19-s − 0.651·22-s + 0.373·23-s + 0.204·24-s + 0.321·26-s + 0.192·27-s + 1.03·29-s + 0.169·31-s + 0.176·32-s − 0.531·33-s − 0.498·34-s + 0.166·36-s − 1.76·37-s + 0.358·38-s + 0.262·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.124483341\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.124483341\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 5.58T + 29T^{2} \) |
| 31 | \( 1 - 0.944T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 5.78T + 53T^{2} \) |
| 59 | \( 1 + 5.97T + 59T^{2} \) |
| 61 | \( 1 - 0.445T + 61T^{2} \) |
| 67 | \( 1 - 1.26T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 7.91T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 0.874T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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
−7.70821045914017013246102453358, −7.30168886903648784897960302259, −6.46762975912836166652199721075, −5.74105281388839072565107368432, −5.00610673377613956261998960819, −4.31825223218995147814513416166, −3.53696078302411505666525004830, −2.74934302333561794743622030683, −2.14128889656692343351598505722, −0.899711084138143284354518122232,
0.899711084138143284354518122232, 2.14128889656692343351598505722, 2.74934302333561794743622030683, 3.53696078302411505666525004830, 4.31825223218995147814513416166, 5.00610673377613956261998960819, 5.74105281388839072565107368432, 6.46762975912836166652199721075, 7.30168886903648784897960302259, 7.70821045914017013246102453358