| L(s) = 1 | − 2-s + 4-s − 1.41·5-s + 1.73·7-s − 8-s + 9-s + 1.41·10-s + 1.93·11-s + 0.517·13-s − 1.73·14-s + 16-s − 17-s − 18-s − 1.41·20-s − 1.93·22-s + 23-s + 1.00·25-s − 0.517·26-s + 1.73·28-s − 32-s + 34-s − 2.44·35-s + 36-s − 0.517·37-s + 1.41·40-s + 1.93·44-s − 1.41·45-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.41·5-s + 1.73·7-s − 8-s + 9-s + 1.41·10-s + 1.93·11-s + 0.517·13-s − 1.73·14-s + 16-s − 17-s − 18-s − 1.41·20-s − 1.93·22-s + 23-s + 1.00·25-s − 0.517·26-s + 1.73·28-s − 32-s + 34-s − 2.44·35-s + 36-s − 0.517·37-s + 1.41·40-s + 1.93·44-s − 1.41·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9517525318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9517525318\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - 1.93T + T^{2} \) |
| 13 | \( 1 - 0.517T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 0.517T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.93T + T^{2} \) |
| 59 | \( 1 - 0.517T + T^{2} \) |
| 61 | \( 1 + 1.93T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.845278736258916560731172936382, −8.193415626700346773617412474766, −7.48010691122988635229865012238, −6.99978690616368086798162909240, −6.21377933429756212488291622829, −4.73695364890531294774643478855, −4.24253395286911504082421748182, −3.37471470469949962521118648341, −1.75928892338640603212995428998, −1.17752680713992252520843253373,
1.17752680713992252520843253373, 1.75928892338640603212995428998, 3.37471470469949962521118648341, 4.24253395286911504082421748182, 4.73695364890531294774643478855, 6.21377933429756212488291622829, 6.99978690616368086798162909240, 7.48010691122988635229865012238, 8.193415626700346773617412474766, 8.845278736258916560731172936382
