L(s) = 1 | + (−1.11 + 1.39i)2-s + (−2.93 + 0.442i)3-s + (−0.263 − 1.15i)4-s + (−0.492 + 0.335i)5-s + (2.64 − 4.58i)6-s + (2.18 + 3.77i)7-s + (−1.31 − 0.632i)8-s + (5.54 − 1.71i)9-s + (0.0795 − 1.06i)10-s + (−0.452 + 1.98i)11-s + (1.28 + 3.26i)12-s + (−0.130 − 1.73i)13-s + (−7.69 − 1.16i)14-s + (1.29 − 1.20i)15-s + (4.47 − 2.15i)16-s + (1.21 + 0.826i)17-s + ⋯ |
L(s) = 1 | + (−0.786 + 0.986i)2-s + (−1.69 + 0.255i)3-s + (−0.131 − 0.576i)4-s + (−0.220 + 0.150i)5-s + (1.08 − 1.87i)6-s + (0.824 + 1.42i)7-s + (−0.464 − 0.223i)8-s + (1.84 − 0.570i)9-s + (0.0251 − 0.335i)10-s + (−0.136 + 0.598i)11-s + (0.370 + 0.943i)12-s + (−0.0361 − 0.482i)13-s + (−2.05 − 0.310i)14-s + (0.334 − 0.310i)15-s + (1.11 − 0.538i)16-s + (0.294 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0909643 + 0.320666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0909643 + 0.320666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-3.81 + 5.32i)T \) |
good | 2 | \( 1 + (1.11 - 1.39i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (2.93 - 0.442i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (0.492 - 0.335i)T + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (-2.18 - 3.77i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.452 - 1.98i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (0.130 + 1.73i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.21 - 0.826i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (2.07 + 0.641i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-1.99 - 1.85i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-7.48 - 1.12i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (0.208 + 0.530i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (-1.98 + 3.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.32 + 1.66i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (2.08 + 9.12i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.441 - 5.88i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (4.60 - 2.21i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (3.57 - 9.10i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (0.506 + 0.156i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-3.91 + 3.63i)T + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (1.05 + 14.0i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (1.82 + 3.16i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 1.54i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (-5.97 + 0.899i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (2.28 - 9.99i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−16.62378239829328268021307521520, −15.48504439234753408853042429297, −15.08873291305536092760529770707, −12.46081300441712166786907905455, −11.77283409742061445358469792704, −10.47371021517231411855563051179, −8.995118290924563013297866809266, −7.52818269411036852534605274152, −6.10959513661904506147022534468, −5.13566128304468705190497729781,
0.951633784206193250715239967742, 4.61456906422842171681142969854, 6.40166147079653704767739797978, 8.039642874087825213679877145370, 10.05856813467542783834335183183, 10.92614207298278185854770008052, 11.49402481838221974567725188215, 12.59385088946712694388547816894, 14.17283677906146033729535455068, 16.16063107877768933584853157012