L(s) = 1 | + 7.07·5-s − 12·7-s − 5.65·11-s − 8i·13-s + 9.89i·17-s − 16i·19-s + 39.5i·23-s + 25.0·25-s + 29.6·29-s − 4·31-s − 84.8·35-s − 30i·37-s + 21.2i·41-s + 8i·43-s − 16.9i·47-s + ⋯ |
L(s) = 1 | + 1.41·5-s − 1.71·7-s − 0.514·11-s − 0.615i·13-s + 0.582i·17-s − 0.842i·19-s + 1.72i·23-s + 1.00·25-s + 1.02·29-s − 0.129·31-s − 2.42·35-s − 0.810i·37-s + 0.517i·41-s + 0.186i·43-s − 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.970913579\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.970913579\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 7.07T + 25T^{2} \) |
| 7 | \( 1 + 12T + 49T^{2} \) |
| 11 | \( 1 + 5.65T + 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 - 9.89iT - 289T^{2} \) |
| 19 | \( 1 + 16iT - 361T^{2} \) |
| 23 | \( 1 - 39.5iT - 529T^{2} \) |
| 29 | \( 1 - 29.6T + 841T^{2} \) |
| 31 | \( 1 + 4T + 961T^{2} \) |
| 37 | \( 1 + 30iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 21.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 79.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 88iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 28.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 80T + 5.32e3T^{2} \) |
| 79 | \( 1 - 100T + 6.24e3T^{2} \) |
| 83 | \( 1 - 130.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 148. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112T + 9.40e3T^{2} \) |
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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
−9.178632052666673902285228715413, −8.060295286956872933018974763478, −7.04600133298256506531994712153, −6.38686954679617545084609626844, −5.74027136689048687725963408808, −5.12594970703650640463497883233, −3.68848482100153404530506415022, −2.92541910291580254220216824144, −2.08015909920273764159468874576, −0.67406195629353260497152062885,
0.74042215696086098366972106701, 2.23300762950498611744592327168, 2.79873633725938034778835438682, 3.89613030431404412011800027140, 5.03681087739045701823947463261, 5.86378901235149043455209746409, 6.53168852275061649227577321325, 6.91571663571941776960399858176, 8.238683355584266398520555149948, 9.056671111360346498160009521805