| L(s) = 1 | − 2-s + 1.80·3-s + 4-s − 1.80·6-s − 3.60·7-s − 8-s + 0.246·9-s − 4.44·11-s + 1.80·12-s + 3.60·14-s + 16-s + 3.15·17-s − 0.246·18-s + 0.356·19-s − 6.49·21-s + 4.44·22-s + 6.49·23-s − 1.80·24-s − 4.96·27-s − 3.60·28-s − 2.89·29-s − 3.82·31-s − 32-s − 8.00·33-s − 3.15·34-s + 0.246·36-s − 7.48·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s − 0.735·6-s − 1.36·7-s − 0.353·8-s + 0.0823·9-s − 1.34·11-s + 0.520·12-s + 0.963·14-s + 0.250·16-s + 0.766·17-s − 0.0582·18-s + 0.0818·19-s − 1.41·21-s + 0.947·22-s + 1.35·23-s − 0.367·24-s − 0.954·27-s − 0.681·28-s − 0.536·29-s − 0.686·31-s − 0.176·32-s − 1.39·33-s − 0.541·34-s + 0.0411·36-s − 1.23·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119474292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119474292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 0.356T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 + 2.03T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 + 8.02T + 89T^{2} \) |
| 97 | \( 1 - 3.95T + 97T^{2} \) |
| 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
−7.84469564859345068424005794936, −7.23619942308462196036483118035, −6.76383528961792590116833879953, −5.64098930841851141810680381034, −5.29989214463769026346543117783, −3.84767953248203422529685378809, −3.18035766043370144421042575406, −2.77812668550935152834710340451, −1.90670011348352381702305438522, −0.51537009904316792917569991822,
0.51537009904316792917569991822, 1.90670011348352381702305438522, 2.77812668550935152834710340451, 3.18035766043370144421042575406, 3.84767953248203422529685378809, 5.29989214463769026346543117783, 5.64098930841851141810680381034, 6.76383528961792590116833879953, 7.23619942308462196036483118035, 7.84469564859345068424005794936
