L(s) = 1 | − 1.56·2-s + 1.44·4-s + 1.80·7-s − 0.695·8-s − 2.81·14-s − 0.356·16-s + 1.94·17-s − 0.445·19-s + 25-s + 2.60·28-s + 1.56·29-s − 1.24·31-s + 1.25·32-s − 3.04·34-s − 1.80·37-s + 0.695·38-s − 0.867·41-s + 1.24·43-s − 1.94·47-s + 2.24·49-s − 1.56·50-s − 1.94·53-s − 1.25·56-s − 2.44·58-s + 1.94·62-s − 1.60·64-s + 2.81·68-s + ⋯ |
L(s) = 1 | − 1.56·2-s + 1.44·4-s + 1.80·7-s − 0.695·8-s − 2.81·14-s − 0.356·16-s + 1.94·17-s − 0.445·19-s + 25-s + 2.60·28-s + 1.56·29-s − 1.24·31-s + 1.25·32-s − 3.04·34-s − 1.80·37-s + 0.695·38-s − 0.867·41-s + 1.24·43-s − 1.94·47-s + 2.24·49-s − 1.56·50-s − 1.94·53-s − 1.25·56-s − 2.44·58-s + 1.94·62-s − 1.60·64-s + 2.81·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7141128749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7141128749\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 2 | \( 1 + 1.56T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.80T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.94T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.56T + T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 + 0.867T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 + 1.94T + T^{2} \) |
| 53 | \( 1 + 1.94T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.56T + T^{2} \) |
| 89 | \( 1 - 0.867T + T^{2} \) |
| 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
−9.156061486191829777328620414598, −8.512269753857074653120670187399, −7.960080656638404662794052927853, −7.45058887344165435370099057293, −6.52311507151261373470107863371, −5.28493490221179802443260702696, −4.67867208623404834743395622367, −3.23709019900996103047607764442, −1.87337273959216694144740636716, −1.19682122534145259993479432148,
1.19682122534145259993479432148, 1.87337273959216694144740636716, 3.23709019900996103047607764442, 4.67867208623404834743395622367, 5.28493490221179802443260702696, 6.52311507151261373470107863371, 7.45058887344165435370099057293, 7.960080656638404662794052927853, 8.512269753857074653120670187399, 9.156061486191829777328620414598