| L(s) = 1 | − 0.347·2-s + 1.87·3-s − 0.879·4-s − 0.652·6-s + 0.652·8-s + 2.53·9-s − 1.65·12-s − 1.53·13-s + 0.652·16-s − 0.879·18-s − 23-s + 1.22·24-s + 0.532·26-s + 2.87·27-s + 0.347·29-s + 0.347·31-s − 0.879·32-s − 2.22·36-s − 2.87·39-s − 1.87·41-s + 0.347·46-s − 1.53·47-s + 1.22·48-s + 49-s + 1.34·52-s − 54-s − 0.120·58-s + ⋯ |
| L(s) = 1 | − 0.347·2-s + 1.87·3-s − 0.879·4-s − 0.652·6-s + 0.652·8-s + 2.53·9-s − 1.65·12-s − 1.53·13-s + 0.652·16-s − 0.879·18-s − 23-s + 1.22·24-s + 0.532·26-s + 2.87·27-s + 0.347·29-s + 0.347·31-s − 0.879·32-s − 2.22·36-s − 2.87·39-s − 1.87·41-s + 0.347·46-s − 1.53·47-s + 1.22·48-s + 49-s + 1.34·52-s − 54-s − 0.120·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.123444418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123444418\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.347T + T^{2} \) |
| 3 | \( 1 - 1.87T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.53T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.347T + T^{2} \) |
| 31 | \( 1 - 0.347T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.53T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.53T + T^{2} \) |
| 73 | \( 1 + 0.347T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−10.30064914588326414884822415972, −9.831735902917001771170123080028, −9.178620434056004068424555058636, −8.294491515645224017743565176338, −7.82280815387553346203126681246, −6.87711148929099842553837276550, −5.01717691935053124454113117108, −4.15527193395974415839074896834, −3.09760388663792715481797239514, −1.89718424583192943844562804149,
1.89718424583192943844562804149, 3.09760388663792715481797239514, 4.15527193395974415839074896834, 5.01717691935053124454113117108, 6.87711148929099842553837276550, 7.82280815387553346203126681246, 8.294491515645224017743565176338, 9.178620434056004068424555058636, 9.831735902917001771170123080028, 10.30064914588326414884822415972
