L(s) = 1 | + 3-s + 9-s − 4·11-s − 2·13-s − 6·17-s + 4·19-s + 8·23-s + 27-s + 6·29-s − 4·33-s + 10·37-s − 2·39-s − 6·41-s − 43-s + 8·47-s − 7·49-s − 6·51-s + 2·53-s + 4·57-s − 4·59-s − 14·61-s + 4·67-s + 8·69-s + 16·71-s − 10·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s + 0.192·27-s + 1.11·29-s − 0.696·33-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 0.152·43-s + 1.16·47-s − 49-s − 0.840·51-s + 0.274·53-s + 0.529·57-s − 0.520·59-s − 1.79·61-s + 0.488·67-s + 0.963·69-s + 1.89·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.296957255\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.296957255\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.46266648708002, −13.90062491167257, −13.48962338871358, −12.94306990972969, −12.70096982550527, −11.93764491276363, −11.29472908838938, −10.92827630023407, −10.30324071680061, −9.793535085619831, −9.281112434644914, −8.682125233157539, −8.311159516843577, −7.455233802392289, −7.350948429102423, −6.573608100403049, −5.989688959443169, −5.070665042881361, −4.831789033049242, −4.227007319780869, −3.232892957987031, −2.793266620556769, −2.352428500452360, −1.408364539714435, −0.5126833804568995,
0.5126833804568995, 1.408364539714435, 2.352428500452360, 2.793266620556769, 3.232892957987031, 4.227007319780869, 4.831789033049242, 5.070665042881361, 5.989688959443169, 6.573608100403049, 7.350948429102423, 7.455233802392289, 8.311159516843577, 8.682125233157539, 9.281112434644914, 9.793535085619831, 10.30324071680061, 10.92827630023407, 11.29472908838938, 11.93764491276363, 12.70096982550527, 12.94306990972969, 13.48962338871358, 13.90062491167257, 14.46266648708002