| L(s) = 1 | − 1.66·2-s − 3.28·3-s + 0.757·4-s + 5.45·6-s − 0.418·7-s + 2.06·8-s + 7.79·9-s − 1.03·11-s − 2.48·12-s − 1.83·13-s + 0.694·14-s − 4.94·16-s + 5.10·17-s − 12.9·18-s + 3.10·19-s + 1.37·21-s + 1.71·22-s − 8.80·23-s − 6.77·24-s + 3.04·26-s − 15.7·27-s − 0.317·28-s − 1.66·29-s − 3.86·31-s + 4.07·32-s + 3.40·33-s − 8.48·34-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 1.89·3-s + 0.378·4-s + 2.22·6-s − 0.158·7-s + 0.729·8-s + 2.59·9-s − 0.312·11-s − 0.718·12-s − 0.507·13-s + 0.185·14-s − 1.23·16-s + 1.23·17-s − 3.05·18-s + 0.713·19-s + 0.299·21-s + 0.366·22-s − 1.83·23-s − 1.38·24-s + 0.596·26-s − 3.03·27-s − 0.0599·28-s − 0.308·29-s − 0.694·31-s + 0.721·32-s + 0.592·33-s − 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2877217949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2877217949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 + 8.80T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 47 | \( 1 + 0.275T + 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−10.10313911990141714581072851623, −9.446863698727471675918433491553, −8.046952750961766400605598709357, −7.46379991447704180459734208560, −6.60599357140918693123416817948, −5.62149008598823282881933881414, −5.00162721642562993326462727137, −3.87690322839047234749426596783, −1.76082413588442134681124457912, −0.54868007108395040135446063144,
0.54868007108395040135446063144, 1.76082413588442134681124457912, 3.87690322839047234749426596783, 5.00162721642562993326462727137, 5.62149008598823282881933881414, 6.60599357140918693123416817948, 7.46379991447704180459734208560, 8.046952750961766400605598709357, 9.446863698727471675918433491553, 10.10313911990141714581072851623
