L(s) = 1 | + 3-s − 2·9-s + 6·11-s + 4·13-s − 2·19-s − 3·23-s − 5·27-s − 3·29-s − 8·31-s + 6·33-s + 4·37-s + 4·39-s + 9·41-s − 7·43-s + 6·53-s − 2·57-s + 6·59-s + 5·61-s + 5·67-s − 3·69-s + 6·71-s + 16·73-s − 2·79-s + 81-s + 3·83-s − 3·87-s − 15·89-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 0.458·19-s − 0.625·23-s − 0.962·27-s − 0.557·29-s − 1.43·31-s + 1.04·33-s + 0.657·37-s + 0.640·39-s + 1.40·41-s − 1.06·43-s + 0.824·53-s − 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.610·67-s − 0.361·69-s + 0.712·71-s + 1.87·73-s − 0.225·79-s + 1/9·81-s + 0.329·83-s − 0.321·87-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.968392334\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.968392334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.59681280339425, −14.95678966364684, −14.56978592883142, −14.13799107234524, −13.65333446328388, −13.00182966868333, −12.45148906865371, −11.72552991535894, −11.22801132934831, −10.96085723613669, −9.955401869215799, −9.321135338901088, −9.040261790498875, −8.367222141534411, −7.984730370992098, −7.033040900802207, −6.551173848211669, −5.879001526173643, −5.417818174102269, −4.187588071941270, −3.863285683719213, −3.322280838520319, −2.290778595851608, −1.665629830898137, −0.7039459031113941,
0.7039459031113941, 1.665629830898137, 2.290778595851608, 3.322280838520319, 3.863285683719213, 4.187588071941270, 5.417818174102269, 5.879001526173643, 6.551173848211669, 7.033040900802207, 7.984730370992098, 8.367222141534411, 9.040261790498875, 9.321135338901088, 9.955401869215799, 10.96085723613669, 11.22801132934831, 11.72552991535894, 12.45148906865371, 13.00182966868333, 13.65333446328388, 14.13799107234524, 14.56978592883142, 14.95678966364684, 15.59681280339425