| L(s) = 1 | − 0.917·2-s − 7.15·4-s + 15.4·5-s − 20.5·7-s + 13.9·8-s − 14.1·10-s + 65.8·11-s + 18.8·14-s + 44.5·16-s − 44.2·17-s + 147.·19-s − 110.·20-s − 60.4·22-s − 53.1·23-s + 114.·25-s + 147.·28-s + 38.6·29-s + 88.3·31-s − 152.·32-s + 40.6·34-s − 318.·35-s + 78.9·37-s − 134.·38-s + 215.·40-s + 354.·41-s − 407.·43-s − 471.·44-s + ⋯ |
| L(s) = 1 | − 0.324·2-s − 0.894·4-s + 1.38·5-s − 1.11·7-s + 0.614·8-s − 0.448·10-s + 1.80·11-s + 0.360·14-s + 0.695·16-s − 0.631·17-s + 1.77·19-s − 1.23·20-s − 0.585·22-s − 0.481·23-s + 0.914·25-s + 0.994·28-s + 0.247·29-s + 0.512·31-s − 0.840·32-s + 0.204·34-s − 1.53·35-s + 0.350·37-s − 0.575·38-s + 0.850·40-s + 1.35·41-s − 1.44·43-s − 1.61·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.963077038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963077038\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.917T + 8T^{2} \) |
| 5 | \( 1 - 15.4T + 125T^{2} \) |
| 7 | \( 1 + 20.5T + 343T^{2} \) |
| 11 | \( 1 - 65.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 38.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 142.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 63.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 287.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 373.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 119.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 554.T + 9.12e5T^{2} \) |
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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
−9.292243908264584726548460305800, −8.705835076587298756081156516706, −7.46030560027935432938942593557, −6.46415046674261737236431880369, −6.01624195774494953301143173375, −5.00064296798156569955848275969, −3.97975426576241859433450064166, −3.06660077971539773066888679367, −1.67125299763538069694510240836, −0.77102746119186337735703738319,
0.77102746119186337735703738319, 1.67125299763538069694510240836, 3.06660077971539773066888679367, 3.97975426576241859433450064166, 5.00064296798156569955848275969, 6.01624195774494953301143173375, 6.46415046674261737236431880369, 7.46030560027935432938942593557, 8.705835076587298756081156516706, 9.292243908264584726548460305800
