L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.691 − 1.90i)3-s + (0.766 + 0.642i)4-s + (1.64 − 1.51i)5-s + 2.02i·6-s + (−2.30 − 0.405i)7-s + (−0.500 − 0.866i)8-s + (−0.835 + 0.701i)9-s + (−2.06 + 0.858i)10-s + (−2.82 − 4.88i)11-s + (0.691 − 1.90i)12-s + (3.98 + 3.34i)13-s + (2.02 + 1.16i)14-s + (−4.01 − 2.08i)15-s + (0.173 + 0.984i)16-s + (−0.926 + 0.777i)17-s + ⋯ |
L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.399 − 1.09i)3-s + (0.383 + 0.321i)4-s + (0.736 − 0.676i)5-s + 0.825i·6-s + (−0.870 − 0.153i)7-s + (−0.176 − 0.306i)8-s + (−0.278 + 0.233i)9-s + (−0.652 + 0.271i)10-s + (−0.850 − 1.47i)11-s + (0.199 − 0.548i)12-s + (1.10 + 0.928i)13-s + (0.541 + 0.312i)14-s + (−1.03 − 0.537i)15-s + (0.0434 + 0.246i)16-s + (−0.224 + 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0747886 - 0.712764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0747886 - 0.712764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-1.64 + 1.51i)T \) |
| 37 | \( 1 + (5.96 + 1.16i)T \) |
good | 3 | \( 1 + (0.691 + 1.90i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (2.30 + 0.405i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.82 + 4.88i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 3.34i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.926 - 0.777i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.34 + 3.69i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.958 - 1.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.455 - 0.263i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.66iT - 31T^{2} \) |
| 41 | \( 1 + (7.68 + 6.44i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 - 0.375T + 43T^{2} \) |
| 47 | \( 1 + (-0.377 - 0.217i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.1 + 2.13i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-14.5 + 2.55i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.21 - 2.64i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.61 - 1.69i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.37 + 1.22i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + (-7.01 - 1.23i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (5.25 + 6.26i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.45 + 0.961i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.929 - 1.60i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.96032350336228225131380943923, −10.10549539426860329241722491639, −8.847945544498735953076468915821, −8.508066426799152610336479072600, −6.98829064919784396370182656474, −6.39464990183819979775737179156, −5.45175049133930683988445343238, −3.50087433063965841211337485490, −1.92976233429560614597919957011, −0.61353406337608710814589099044,
2.29051318507899674355476658397, 3.73056342888438075007120627757, 5.22270339886848881745390801418, 6.03852315025471441665287320543, 7.03217642764315566269902914536, 8.200294800155448219066139628678, 9.499947301459844150448402706032, 10.12655707758944326309416077094, 10.36913754118098057111292310923, 11.39038018489308536053441125853