| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s + 5-s − 4·6-s − 7-s + 9-s − 2·10-s + 4·12-s + 13-s + 2·14-s + 2·15-s − 4·16-s − 17-s − 2·18-s − 2·19-s + 2·20-s − 2·21-s − 23-s + 25-s − 2·26-s − 4·27-s − 2·28-s − 2·29-s − 4·30-s − 5·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s + 0.447·5-s − 1.63·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s + 1.15·12-s + 0.277·13-s + 0.534·14-s + 0.516·15-s − 16-s − 0.242·17-s − 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.436·21-s − 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.377·28-s − 0.371·29-s − 0.730·30-s − 0.898·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71995 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71995 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.42548932296480, −13.83557801012756, −13.37349958451342, −13.07926467932050, −12.40972399058520, −11.64088444745209, −11.18668493929829, −10.52202094365308, −10.25196040091109, −9.500428809536041, −9.305478621331908, −8.817615269634796, −8.379125158588564, −7.915937653038597, −7.371382630120029, −6.838342008693527, −6.285382799325321, −5.588292898468473, −4.900381159537423, −3.983234889298377, −3.614399599031100, −2.743403096607977, −2.195320678942217, −1.779292000368640, −0.8810508898606011, 0,
0.8810508898606011, 1.779292000368640, 2.195320678942217, 2.743403096607977, 3.614399599031100, 3.983234889298377, 4.900381159537423, 5.588292898468473, 6.285382799325321, 6.838342008693527, 7.371382630120029, 7.915937653038597, 8.379125158588564, 8.817615269634796, 9.305478621331908, 9.500428809536041, 10.25196040091109, 10.52202094365308, 11.18668493929829, 11.64088444745209, 12.40972399058520, 13.07926467932050, 13.37349958451342, 13.83557801012756, 14.42548932296480
