L(s) = 1 | + 3.37i·3-s + 2.52i·5-s − 3.31·7-s − 8.37·9-s − 2.37i·11-s + 5.84i·13-s − 8.51·15-s + 5·17-s − i·19-s − 11.1i·21-s − 0.792·23-s − 1.37·25-s − 18.1i·27-s − 2.67i·29-s − 3.46·31-s + ⋯ |
L(s) = 1 | + 1.94i·3-s + 1.12i·5-s − 1.25·7-s − 2.79·9-s − 0.715i·11-s + 1.61i·13-s − 2.19·15-s + 1.21·17-s − 0.229i·19-s − 2.44i·21-s − 0.165·23-s − 0.274·25-s − 3.48i·27-s − 0.496i·29-s − 0.622·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7560444087\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7560444087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 - 3.37iT - 3T^{2} \) |
| 5 | \( 1 - 2.52iT - 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 13 | \( 1 - 5.84iT - 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 0.792T + 23T^{2} \) |
| 29 | \( 1 + 2.67iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 10.0iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 - 0.644T + 47T^{2} \) |
| 53 | \( 1 - 6.13iT - 53T^{2} \) |
| 59 | \( 1 + 1.37iT - 59T^{2} \) |
| 61 | \( 1 + 14.5iT - 61T^{2} \) |
| 67 | \( 1 + 2.62iT - 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 0.294T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−10.08896169482734412732667088776, −9.678536998112418118376831835959, −9.097462480631607931074309237382, −8.069636940753382475288799253180, −6.66845416519706802794804366551, −6.17339158475382890706296268520, −5.14746427976961150171856336284, −4.02578017180304037392457850052, −3.38000801168883427943071228047, −2.74784087615990220503425551302,
0.33758132492753181000324643856, 1.29382277824264498954001332178, 2.57667518255882584701265560532, 3.55694211544024947910049892385, 5.43997765691192411642242726781, 5.68898952834443705400726762716, 6.82395277533586847794729196559, 7.50044481868229958723669777156, 8.158511881012632733825600450330, 8.939797706059125665078396854096