L(s) = 1 | − 2.61i·2-s − i·3-s − 4.85·4-s − 2.61·6-s − 1.38i·7-s + 7.47i·8-s − 9-s − 6.23·11-s + 4.85i·12-s + 3i·13-s − 3.61·14-s + 9.85·16-s − 3.38i·17-s + 2.61i·18-s + 19-s + ⋯ |
L(s) = 1 | − 1.85i·2-s − 0.577i·3-s − 2.42·4-s − 1.06·6-s − 0.522i·7-s + 2.64i·8-s − 0.333·9-s − 1.88·11-s + 1.40i·12-s + 0.832i·13-s − 0.966·14-s + 2.46·16-s − 0.820i·17-s + 0.617i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.440880 + 0.440880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.440880 + 0.440880i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.61iT - 2T^{2} \) |
| 7 | \( 1 + 1.38iT - 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 3.38iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 6.70iT - 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 + 2.38iT - 43T^{2} \) |
| 47 | \( 1 + 9.18iT - 47T^{2} \) |
| 53 | \( 1 + 3.61iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 1.14T + 71T^{2} \) |
| 73 | \( 1 - 9.14iT - 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 + 8.32iT - 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 9.56iT - 97T^{2} \) |
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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
−10.66793099325938473844076446634, −10.22834360530145057800568942106, −9.082207273968074586678402045067, −8.184914313639160075781557582664, −7.07345285312415941972646366686, −5.34271061012013849433052554493, −4.39796620724941943687918519115, −3.00104358916607971518895108346, −2.10509816619017424405030812614, −0.41033307837234712293814510122,
3.19080851116727452408226245903, 4.74089049241485198751686215205, 5.44793070893091285586227159966, 6.13840970801601055849563194524, 7.62510335338059115906055693201, 8.033489904254586694243826413676, 9.011581060394639498720621405464, 9.961011177252827251532323215365, 10.79705193876498272084798377729, 12.41088839020270263090934531255