L(s) = 1 | + (0.684 − 1.18i)3-s + (0.780 − 1.35i)5-s + (0.561 + 0.972i)9-s + 3.50·11-s + (−0.219 − 0.379i)13-s + (−1.06 − 1.85i)15-s + (0.219 − 0.379i)17-s + (3.12 − 3.03i)19-s + (−2.43 − 4.22i)23-s + (1.28 + 2.21i)25-s + 5.64·27-s + (2.34 + 4.05i)29-s − 2.73·31-s + (2.40 − 4.16i)33-s + 1.12·37-s + ⋯ |
L(s) = 1 | + (0.395 − 0.684i)3-s + (0.349 − 0.604i)5-s + (0.187 + 0.324i)9-s + 1.05·11-s + (−0.0608 − 0.105i)13-s + (−0.276 − 0.478i)15-s + (0.0531 − 0.0920i)17-s + (0.716 − 0.697i)19-s + (−0.508 − 0.881i)23-s + (0.256 + 0.443i)25-s + 1.08·27-s + (0.434 + 0.753i)29-s − 0.492·31-s + (0.418 − 0.724i)33-s + 0.184·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.205731992\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.205731992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.12 + 3.03i)T \) |
good | 3 | \( 1 + (-0.684 + 1.18i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.780 + 1.35i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 + (0.219 + 0.379i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.219 + 0.379i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.43 + 4.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.34 - 4.05i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 6.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.06 + 1.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.219 - 0.379i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.684 - 1.18i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.21 + 2.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.93 + 12.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.43 + 4.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.623 - 1.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.06 - 1.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + (-5.34 - 9.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.62 - 9.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.335851808746206440899789534861, −8.841067737985374787969015900894, −7.961122115970662462214730632149, −7.12059590903340567410620637102, −6.43676982825889783400934877645, −5.29160574015009153730281018392, −4.50425817492623225662538506380, −3.24329474933190390791436004756, −2.03186209115015307330478902128, −1.06105465768708348292029313173,
1.43799690357337026626292481918, 2.87163677589126435014264446968, 3.73971789699617590188181979707, 4.48860600329422414369132210504, 5.80294294831841403326153124633, 6.49192177094923855637419405292, 7.38856699768573859927219742236, 8.373752377454664854422391203693, 9.325861849812841966763612807487, 9.773095968645352311655491175432