| L(s) = 1 | − 0.414·2-s − 3-s − 1.82·4-s + 0.414·6-s + 7-s + 1.58·8-s + 9-s + 2·11-s + 1.82·12-s − 13-s − 0.414·14-s + 3·16-s + 4.82·17-s − 0.414·18-s + 5.65·19-s − 21-s − 0.828·22-s + 6.82·23-s − 1.58·24-s − 5·25-s + 0.414·26-s − 27-s − 1.82·28-s + 7.65·29-s − 9.65·31-s − 4.41·32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.577·3-s − 0.914·4-s + 0.169·6-s + 0.377·7-s + 0.560·8-s + 0.333·9-s + 0.603·11-s + 0.527·12-s − 0.277·13-s − 0.110·14-s + 0.750·16-s + 1.17·17-s − 0.0976·18-s + 1.29·19-s − 0.218·21-s − 0.176·22-s + 1.42·23-s − 0.323·24-s − 25-s + 0.0812·26-s − 0.192·27-s − 0.345·28-s + 1.42·29-s − 1.73·31-s − 0.780·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8306530505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8306530505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−11.91778421432812333688761023369, −10.93021901693832264467984756681, −9.868129868172996668906535224299, −9.226081946387546687606087650199, −8.035619877352594052656783644743, −7.13600036933454380613210868054, −5.62715207324924267832557476789, −4.85387741242917259082489195629, −3.54290927457861224230656796089, −1.13885815283392249125382284398,
1.13885815283392249125382284398, 3.54290927457861224230656796089, 4.85387741242917259082489195629, 5.62715207324924267832557476789, 7.13600036933454380613210868054, 8.035619877352594052656783644743, 9.226081946387546687606087650199, 9.868129868172996668906535224299, 10.93021901693832264467984756681, 11.91778421432812333688761023369
