L(s) = 1 | + (1.5 + 0.866i)7-s + (−0.5 − 0.866i)13-s + 1.73i·19-s + (0.5 − 0.866i)25-s − 37-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (1.5 − 0.866i)67-s + 73-s + (−1.5 − 0.866i)79-s − 1.73i·91-s + (−0.5 + 0.866i)97-s + (−1.5 + 0.866i)103-s − 2·109-s + ⋯ |
L(s) = 1 | + (1.5 + 0.866i)7-s + (−0.5 − 0.866i)13-s + 1.73i·19-s + (0.5 − 0.866i)25-s − 37-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (1.5 − 0.866i)67-s + 73-s + (−1.5 − 0.866i)79-s − 1.73i·91-s + (−0.5 + 0.866i)97-s + (−1.5 + 0.866i)103-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.227530840\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227530840\) |
\(L(1)\) |
|
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 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 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 | \( 1 + T + T^{2} \) |
| 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 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.996510212850216340216091384203, −8.938286671564841834798194742013, −8.106568627095995396899443865739, −7.84065385959192551477496001007, −6.54017162726213376600485443005, −5.49121248257260812318924678306, −5.04319538511313213439944940436, −3.88679375911641423821655249687, −2.59408595753712814717891629242, −1.58358352662493684964319040722,
1.30334691944177848365706450745, 2.48166398398396490622656458135, 3.93033597963368283782241038190, 4.74068328645185722852966382623, 5.33514938889448420725420702813, 6.92202543814077441186939196323, 7.16106758434397464540285792914, 8.222003787667362122764573021420, 8.921861179008962435575415001843, 9.791902615239521127313433528469