L(s) = 1 | + (0.564 + 1.29i)2-s + (−1.36 + 1.46i)4-s + (2.22 + 0.173i)5-s + 3.43i·7-s + (−2.66 − 0.943i)8-s + (1.03 + 2.98i)10-s + 4.54i·11-s + 1.84·13-s + (−4.45 + 1.93i)14-s + (−0.281 − 3.99i)16-s − 0.380i·17-s − 1.23i·19-s + (−3.29 + 3.02i)20-s + (−5.89 + 2.56i)22-s − 5.35i·23-s + ⋯ |
L(s) = 1 | + (0.398 + 0.917i)2-s + (−0.681 + 0.731i)4-s + (0.996 + 0.0774i)5-s + 1.29i·7-s + (−0.942 − 0.333i)8-s + (0.326 + 0.945i)10-s + 1.37i·11-s + 0.510·13-s + (−1.18 + 0.517i)14-s + (−0.0702 − 0.997i)16-s − 0.0923i·17-s − 0.282i·19-s + (−0.736 + 0.676i)20-s + (−1.25 + 0.546i)22-s − 1.11i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.003931403\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.003931403\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.564 - 1.29i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.22 - 0.173i)T \) |
good | 7 | \( 1 - 3.43iT - 7T^{2} \) |
| 11 | \( 1 - 4.54iT - 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 + 0.380iT - 17T^{2} \) |
| 19 | \( 1 + 1.23iT - 19T^{2} \) |
| 23 | \( 1 + 5.35iT - 23T^{2} \) |
| 29 | \( 1 - 3.17iT - 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 + 6.60T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.34T + 43T^{2} \) |
| 47 | \( 1 - 9.34iT - 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 - 3.34iT - 59T^{2} \) |
| 61 | \( 1 - 7.74iT - 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 13.1iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 0.240T + 83T^{2} \) |
| 89 | \( 1 + 5.46T + 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−9.957407423687813205020784901770, −9.085650922501519939473333522723, −8.804678125969004355748823827600, −7.57568412052679253202654030953, −6.71957712535277045978312682116, −5.98632058355549021923130507926, −5.25348608265598381314444013030, −4.46922397190181220150618141303, −2.99949470291049401828671347305, −1.98862608515549418211671561854,
0.801918586061035010074760596472, 1.87229517222004784781014108920, 3.32202932742600389736409869164, 3.91239729888669125885156306429, 5.22190892666600204680688949341, 5.86512239399782840104310452416, 6.78310211889074064860201383726, 8.051154546536896278525183829097, 8.986451494813204874629814139273, 9.698595280677114148543499294737