| L(s) = 1 | + 3-s + 9-s + 4·17-s − 2·19-s − 23-s + 27-s − 6·29-s − 2·37-s + 6·41-s − 4·43-s − 8·47-s − 7·49-s + 4·51-s − 8·53-s − 2·57-s − 6·59-s + 10·61-s + 4·67-s − 69-s − 2·71-s + 16·73-s + 14·79-s + 81-s + 4·83-s − 6·87-s − 10·89-s + 10·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.970·17-s − 0.458·19-s − 0.208·23-s + 0.192·27-s − 1.11·29-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.560·51-s − 1.09·53-s − 0.264·57-s − 0.781·59-s + 1.28·61-s + 0.488·67-s − 0.120·69-s − 0.237·71-s + 1.87·73-s + 1.57·79-s + 1/9·81-s + 0.439·83-s − 0.643·87-s − 1.05·89-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.48038308739800, −14.79832458745384, −14.54110464353542, −13.99435007750997, −13.40089314863746, −12.78254176421942, −12.51276693485799, −11.74900351088529, −11.17985722706206, −10.67128838717126, −9.921738679107968, −9.566524917832887, −9.027377934940563, −8.217823745231214, −7.932846220179798, −7.335119307489699, −6.557273651976786, −6.113764529794184, −5.209664217307466, −4.821007334054126, −3.763831715604570, −3.548164293719193, −2.639831468597890, −1.933719342694572, −1.181729163661641, 0,
1.181729163661641, 1.933719342694572, 2.639831468597890, 3.548164293719193, 3.763831715604570, 4.821007334054126, 5.209664217307466, 6.113764529794184, 6.557273651976786, 7.335119307489699, 7.932846220179798, 8.217823745231214, 9.027377934940563, 9.566524917832887, 9.921738679107968, 10.67128838717126, 11.17985722706206, 11.74900351088529, 12.51276693485799, 12.78254176421942, 13.40089314863746, 13.99435007750997, 14.54110464353542, 14.79832458745384, 15.48038308739800
