| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 0.445·5-s − 7-s − 0.554·8-s + 9-s + 0.554·10-s + 1.80·13-s − 1.24·14-s − 1.24·16-s − 1.24·17-s + 1.24·18-s + 0.246·20-s − 1.80·23-s − 0.801·25-s + 2.24·26-s − 0.554·28-s − 1.24·31-s − 0.999·32-s − 1.55·34-s − 0.445·35-s + 0.554·36-s − 0.445·37-s − 0.246·40-s + 0.445·45-s − 2.24·46-s + 1.80·47-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 0.445·5-s − 7-s − 0.554·8-s + 9-s + 0.554·10-s + 1.80·13-s − 1.24·14-s − 1.24·16-s − 1.24·17-s + 1.24·18-s + 0.246·20-s − 1.80·23-s − 0.801·25-s + 2.24·26-s − 0.554·28-s − 1.24·31-s − 0.999·32-s − 1.55·34-s − 0.445·35-s + 0.554·36-s − 0.445·37-s − 0.246·40-s + 0.445·45-s − 2.24·46-s + 1.80·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.513936964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513936964\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.445T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.80T + T^{2} \) |
| 17 | \( 1 + 1.24T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 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
−11.27844432157517089249710715628, −10.30094523872363655365864930350, −9.405071121813400764730642115080, −8.572194452010895270314546746972, −7.04276434257193076895792618328, −6.21566270010054295059896466528, −5.64574706609525860024357507646, −4.09889796273287446407773117432, −3.73833709041184025770710597953, −2.13566220646188944582769972128,
2.13566220646188944582769972128, 3.73833709041184025770710597953, 4.09889796273287446407773117432, 5.64574706609525860024357507646, 6.21566270010054295059896466528, 7.04276434257193076895792618328, 8.572194452010895270314546746972, 9.405071121813400764730642115080, 10.30094523872363655365864930350, 11.27844432157517089249710715628
