L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s − i·7-s + (−0.707 + 0.707i)8-s + 1.93·11-s + (−0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s + (0.499 − 1.86i)22-s + 1.41·23-s − 25-s + (−0.499 + 0.866i)28-s − 1.41i·29-s + (0.965 − 0.258i)32-s − 1.73·37-s − i·43-s + (−1.67 − 0.965i)44-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s − i·7-s + (−0.707 + 0.707i)8-s + 1.93·11-s + (−0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s + (0.499 − 1.86i)22-s + 1.41·23-s − 25-s + (−0.499 + 0.866i)28-s − 1.41i·29-s + (0.965 − 0.258i)32-s − 1.73·37-s − i·43-s + (−1.67 − 0.965i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.309726718\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.309726718\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.93T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.73T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.517iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 - 0.517T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.141884711816720741104122800239, −8.470144453855161664759413934214, −7.32552820905862677192465307662, −6.60120879230057414738653959337, −5.70618522351145562200286207495, −4.60696002890717912375442354551, −3.95042565288546613742079792343, −3.34168721974928137322403751866, −1.93393750211832871243819754605, −0.961686503587871155622159828237,
1.53780866993924077354602708405, 3.12002806879352795613672669603, 3.86283943186963585835606029353, 4.90873460090732039515494767614, 5.55673572662266305972009463859, 6.55366713338185043424862343923, 6.82526861683469798058967530452, 7.918206742953217995804092225153, 8.840237176001285805919434402645, 9.086769072817375970327685975123