L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·11-s − 6·13-s − 16-s + 6·17-s + 8·19-s + 20-s − 4·22-s − 8·23-s + 25-s − 6·26-s − 6·29-s − 4·31-s + 5·32-s + 6·34-s + 2·37-s + 8·38-s + 3·40-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 7·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.20·11-s − 1.66·13-s − 1/4·16-s + 1.45·17-s + 1.83·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s + 0.328·37-s + 1.29·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.259639989\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259639989\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.146202063001343333418383946177, −7.72599231559197505315957318438, −7.19752949399766977110725961254, −5.76330938702987298190551180645, −5.43133576226304629847483472279, −4.79928131936580810015551866294, −3.83026869442654314236078087892, −3.17039510608884563762840221305, −2.28305339484288399123535724073, −0.55475894028473866688325407921,
0.55475894028473866688325407921, 2.28305339484288399123535724073, 3.17039510608884563762840221305, 3.83026869442654314236078087892, 4.79928131936580810015551866294, 5.43133576226304629847483472279, 5.76330938702987298190551180645, 7.19752949399766977110725961254, 7.72599231559197505315957318438, 8.146202063001343333418383946177