| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s − 2·11-s + 12-s + 6·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s + 4·21-s + 2·22-s − 5·23-s − 24-s − 6·26-s + 27-s + 4·28-s + 10·31-s − 32-s − 2·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.426·22-s − 1.04·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.79·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.147351202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147351202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.605637047293088628845868985582, −8.188287204909209328453849372785, −7.80740081157824912500538828494, −6.79463681343707149067236637719, −5.84084108193046638467245630585, −4.99585376514666012367397104697, −3.99225334470857281631445650152, −3.01308414951224896614167330334, −1.88247361842834831005657001491, −1.11251462700235043615266083539,
1.11251462700235043615266083539, 1.88247361842834831005657001491, 3.01308414951224896614167330334, 3.99225334470857281631445650152, 4.99585376514666012367397104697, 5.84084108193046638467245630585, 6.79463681343707149067236637719, 7.80740081157824912500538828494, 8.188287204909209328453849372785, 8.605637047293088628845868985582
