L(s) = 1 | − 3.09·5-s + i·7-s + 0.646i·11-s + 4.37i·13-s − 0.295i·17-s − 5.29·19-s + 3.94·23-s + 4.58·25-s + 5.03·29-s + 9.16i·31-s − 3.09i·35-s + 2.55i·37-s − 10.4i·41-s − 1.63·43-s + 5.65·47-s + ⋯ |
L(s) = 1 | − 1.38·5-s + 0.377i·7-s + 0.194i·11-s + 1.21i·13-s − 0.0715i·17-s − 1.21·19-s + 0.823·23-s + 0.916·25-s + 0.934·29-s + 1.64i·31-s − 0.523i·35-s + 0.419i·37-s − 1.62i·41-s − 0.249·43-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.003310684799\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003310684799\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 3.09T + 5T^{2} \) |
| 11 | \( 1 - 0.646iT - 11T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 + 0.295iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 - 2.55iT - 37T^{2} \) |
| 41 | \( 1 + 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 1.11T + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 16.7iT - 79T^{2} \) |
| 83 | \( 1 + 8.50iT - 83T^{2} \) |
| 89 | \( 1 - 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.350166158273444354995029340817, −7.26363856733779815096814507987, −6.92532667042511413754565231221, −6.05506944748167140759489139907, −4.87201149565516689555414064940, −4.40460669796012399297531350781, −3.59180142042786400488597987860, −2.68160593886245521990038215729, −1.51827853615113300400126193082, −0.00115759301572620047819726424,
0.990617429914980236547881348023, 2.58257664872019655946553144150, 3.38422774411090623132754841651, 4.17883094253160939990629994935, 4.76455012295913024547826006264, 5.82087784163664131367487222901, 6.61273514329047475078856582664, 7.42290294826871102545599110855, 8.090330151342734176256746644528, 8.366846091738233565932951490331