| L(s) = 1 | − 3-s − 4·7-s + 9-s − 4·11-s − 2·13-s + 6·17-s + 19-s + 4·21-s − 4·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s − 10·37-s + 2·39-s + 2·41-s + 4·47-s + 9·49-s − 6·51-s − 2·53-s − 57-s − 12·59-s − 2·61-s − 4·63-s − 4·67-s + 4·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.229·19-s + 0.872·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.583·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s − 0.132·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s − 0.488·67-s + 0.481·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.72459004527026, −16.09058896603410, −15.66156535838854, −15.41310250708560, −14.30430282881790, −13.83516789509485, −13.29380798663411, −12.46569084741741, −12.30394185028904, −11.81565798307224, −10.69966815276558, −10.27631316119146, −9.939898910230539, −9.353274329805021, −8.421743605633587, −7.727535585087895, −7.237215747963065, −6.345316735899821, −6.028958502859256, −5.196724598123839, −4.677312664455673, −3.549034555925662, −3.066735105885855, −2.255779237560669, −0.9032030126911798, 0,
0.9032030126911798, 2.255779237560669, 3.066735105885855, 3.549034555925662, 4.677312664455673, 5.196724598123839, 6.028958502859256, 6.345316735899821, 7.237215747963065, 7.727535585087895, 8.421743605633587, 9.353274329805021, 9.939898910230539, 10.27631316119146, 10.69966815276558, 11.81565798307224, 12.30394185028904, 12.46569084741741, 13.29380798663411, 13.83516789509485, 14.30430282881790, 15.41310250708560, 15.66156535838854, 16.09058896603410, 16.72459004527026
