L(s) = 1 | + 0.115·2-s − 3.28·3-s − 1.98·4-s − 1.31·5-s − 0.379·6-s + 3.19·7-s − 0.461·8-s + 7.77·9-s − 0.151·10-s + 1.38·11-s + 6.52·12-s − 5.87·13-s + 0.369·14-s + 4.30·15-s + 3.91·16-s + 5.28·17-s + 0.899·18-s + 4.99·19-s + 2.60·20-s − 10.4·21-s + 0.160·22-s + 3.07·23-s + 1.51·24-s − 3.28·25-s − 0.679·26-s − 15.6·27-s − 6.35·28-s + ⋯ |
L(s) = 1 | + 0.0817·2-s − 1.89·3-s − 0.993·4-s − 0.586·5-s − 0.155·6-s + 1.20·7-s − 0.163·8-s + 2.59·9-s − 0.0479·10-s + 0.419·11-s + 1.88·12-s − 1.62·13-s + 0.0988·14-s + 1.11·15-s + 0.979·16-s + 1.28·17-s + 0.212·18-s + 1.14·19-s + 0.582·20-s − 2.28·21-s + 0.0342·22-s + 0.640·23-s + 0.309·24-s − 0.656·25-s − 0.133·26-s − 3.01·27-s − 1.20·28-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
\(L(1)\) |
\(\approx\) |
\(0.5623544471\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5623544471\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.115T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 0.672T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 - 0.970T + 41T^{2} \) |
| 43 | \( 1 - 7.93T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 8.45T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 - 0.717T + 61T^{2} \) |
| 67 | \( 1 - 8.81T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 1.18T + 97T^{2} \) |
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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
−12.03423662288421035556878010200, −11.45491668813257385004805761554, −10.30604059905496515022833906730, −9.546796007397578905652753495475, −7.909811382413088058571791126211, −7.16374561311484867983482135791, −5.48151733873869996502950363066, −5.06068892700560168139239033355, −4.08777130419848274595431752726, −0.925914816352511531387618261683,
0.925914816352511531387618261683, 4.08777130419848274595431752726, 5.06068892700560168139239033355, 5.48151733873869996502950363066, 7.16374561311484867983482135791, 7.909811382413088058571791126211, 9.546796007397578905652753495475, 10.30604059905496515022833906730, 11.45491668813257385004805761554, 12.03423662288421035556878010200