L(s) = 1 | + (−0.866 − 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.866 + 0.5i)25-s + (−1 − i)29-s + (−0.366 − 1.36i)37-s + (−1 + i)43-s + (−1.36 − 0.366i)53-s + (−0.366 + 1.36i)67-s − 2i·71-s + (0.499 + 0.866i)81-s + (0.999 + i)99-s + (−0.366 − 1.36i)107-s + (0.366 − 1.36i)109-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.866 + 0.5i)25-s + (−1 − i)29-s + (−0.366 − 1.36i)37-s + (−1 + i)43-s + (−1.36 − 0.366i)53-s + (−0.366 + 1.36i)67-s − 2i·71-s + (0.499 + 0.866i)81-s + (0.999 + i)99-s + (−0.366 − 1.36i)107-s + (0.366 − 1.36i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3199560453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3199560453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.427996418459638467057432088843, −7.900512930318205910375536823986, −7.16738523041969537860649366702, −6.02657825756519963457876702069, −5.67309401242918886976098771410, −4.77054521313406216804833145599, −3.66167939576671648536203342340, −2.93313081480110696234522764150, −1.94836587209858076261051358068, −0.17520951856914355520551977233,
1.82753241776595860201490764989, 2.71081471819729334289083109492, 3.55142938201465429828074858978, 4.79171495365004877020756195307, 5.29139649609443262546872212072, 6.06959998955496526407695458299, 7.03404360216599192855688667931, 7.83994779585751711375043450142, 8.293296188471047849945881907972, 9.114815122480079394975096333348