L(s) = 1 | + (1.57 + 0.714i)3-s + (−1.31 + 1.31i)5-s + 7-s + (1.97 + 2.25i)9-s + (−1.23 − 1.23i)11-s + (4.57 − 4.57i)13-s + (−3.02 + 1.13i)15-s + 5.82i·17-s + (2.27 + 2.27i)19-s + (1.57 + 0.714i)21-s − 2.63i·23-s + 1.51i·25-s + (1.50 + 4.97i)27-s + (4.30 + 4.30i)29-s − 5.81i·31-s + ⋯ |
L(s) = 1 | + (0.910 + 0.412i)3-s + (−0.590 + 0.590i)5-s + 0.377·7-s + (0.659 + 0.751i)9-s + (−0.372 − 0.372i)11-s + (1.26 − 1.26i)13-s + (−0.781 + 0.294i)15-s + 1.41i·17-s + (0.521 + 0.521i)19-s + (0.344 + 0.155i)21-s − 0.549i·23-s + 0.303i·25-s + (0.290 + 0.956i)27-s + (0.799 + 0.799i)29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.262568201\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.262568201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 - 0.714i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (1.31 - 1.31i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.23 + 1.23i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.82iT - 17T^{2} \) |
| 19 | \( 1 + (-2.27 - 2.27i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.63iT - 23T^{2} \) |
| 29 | \( 1 + (-4.30 - 4.30i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.81iT - 31T^{2} \) |
| 37 | \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 + (2.12 - 2.12i)T - 43iT^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (-0.573 + 0.573i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.42 - 8.42i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.80 - 6.80i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.90 + 1.90i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 0.516iT - 73T^{2} \) |
| 79 | \( 1 - 3.10iT - 79T^{2} \) |
| 83 | \( 1 + (-2.91 + 2.91i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 - 8.09T + 97T^{2} \) |
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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
−9.872086435607186041351362266083, −8.644966600116997805004309522418, −8.146674049091744581412661294148, −7.72627326405958924634019318239, −6.48768876509629992671113449239, −5.57333120355412657142300474002, −4.41628233107678471008780350107, −3.47180450531509015598254551900, −2.97163419744072237241052544740, −1.43483681019697165709700340872,
0.960095549713995977401335590164, 2.14830200605394870810647119947, 3.32429929492023765118319699614, 4.29252777721519369995436666212, 5.01427726763129470218565385822, 6.43509544453475216654288578412, 7.17269531380470865830201000499, 7.961916443616945933966350738630, 8.626866823734521249409237833773, 9.258128662158001889987651767230