L(s) = 1 | + (1 − 1.73i)13-s + (0.5 + 0.866i)25-s + 2·37-s + (−0.5 + 0.866i)49-s + (−1 − 1.73i)61-s − 2·73-s + (1 + 1.73i)97-s − 2·109-s + ⋯ |
L(s) = 1 | + (1 − 1.73i)13-s + (0.5 + 0.866i)25-s + 2·37-s + (−0.5 + 0.866i)49-s + (−1 − 1.73i)61-s − 2·73-s + (1 + 1.73i)97-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.116706715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116706715\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 2T + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.799943370483185191917632267189, −8.998251954515816010525002972349, −8.089965652763877437246867911823, −7.56665921137478423816141511945, −6.35948809764005141177484987876, −5.70909325345432036401935897155, −4.76514537377726367324841783769, −3.59016694608725740294725653465, −2.77700903918408898284677472720, −1.16768116849375470807889173288,
1.46977545229909853101001693791, 2.71058323338231288014984230798, 3.99686914297187884985064830764, 4.60813037763135039422912437780, 5.92012251779772620727959588417, 6.53338037649943581419368298978, 7.40164564533685454789307351100, 8.436774389104417240398597665973, 9.024673114278215893458209243182, 9.826207567758078568599489847543