| L(s) = 1 | − 1.35·2-s − 2.45·3-s − 0.160·4-s + 2.74·5-s + 3.32·6-s − 0.283·7-s + 2.93·8-s + 3.00·9-s − 3.71·10-s − 4.12·11-s + 0.393·12-s + 0.0271·13-s + 0.384·14-s − 6.71·15-s − 3.65·16-s − 1.28·17-s − 4.07·18-s − 5.72·19-s − 0.440·20-s + 0.695·21-s + 5.59·22-s − 5.97·23-s − 7.18·24-s + 2.51·25-s − 0.0368·26-s − 0.0132·27-s + 0.0455·28-s + ⋯ |
| L(s) = 1 | − 0.959·2-s − 1.41·3-s − 0.0802·4-s + 1.22·5-s + 1.35·6-s − 0.107·7-s + 1.03·8-s + 1.00·9-s − 1.17·10-s − 1.24·11-s + 0.113·12-s + 0.00754·13-s + 0.102·14-s − 1.73·15-s − 0.913·16-s − 0.312·17-s − 0.960·18-s − 1.31·19-s − 0.0984·20-s + 0.151·21-s + 1.19·22-s − 1.24·23-s − 1.46·24-s + 0.503·25-s − 0.00723·26-s − 0.00254·27-s + 0.00860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 + 0.283T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 - 0.0271T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 + 9.94T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 6.69T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 - 0.746T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−11.25003837187308958465838810398, −10.43204801176809781890246576973, −10.02501184550850767951587913083, −8.903394041707753737205291346411, −7.74254879183894282901852152874, −6.41021242033236520191815438528, −5.60925018361799856285197653623, −4.59652166428247929016248239880, −1.97523602169959753657160610366, 0,
1.97523602169959753657160610366, 4.59652166428247929016248239880, 5.60925018361799856285197653623, 6.41021242033236520191815438528, 7.74254879183894282901852152874, 8.903394041707753737205291346411, 10.02501184550850767951587913083, 10.43204801176809781890246576973, 11.25003837187308958465838810398
