| L(s) = 1 | + 3-s + 9-s + 11-s + 2·13-s − 3·17-s + 7·19-s + 3·23-s + 27-s − 3·29-s + 4·31-s + 33-s + 10·37-s + 2·39-s + 7·43-s − 3·51-s − 9·53-s + 7·57-s − 3·59-s + 61-s − 2·67-s + 3·69-s − 12·71-s − 4·73-s + 2·79-s + 81-s − 3·83-s − 3·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.727·17-s + 1.60·19-s + 0.625·23-s + 0.192·27-s − 0.557·29-s + 0.718·31-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 1.06·43-s − 0.420·51-s − 1.23·53-s + 0.927·57-s − 0.390·59-s + 0.128·61-s − 0.244·67-s + 0.361·69-s − 1.42·71-s − 0.468·73-s + 0.225·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.46359526173336, −13.23222340730621, −12.58309433356499, −12.12636694976215, −11.55729185263686, −11.06221265645909, −10.84862934910401, −9.982061306539794, −9.598644304658496, −9.226018719475681, −8.783659872055481, −8.149685394945403, −7.764452565728892, −7.217794580925693, −6.782618541225109, −6.108947770443638, −5.711517716457019, −5.030578702273187, −4.388891974949197, −4.053326890961483, −3.277313270270915, −2.861572970410884, −2.336304583280494, −1.310155225334694, −1.154180628876302, 0,
1.154180628876302, 1.310155225334694, 2.336304583280494, 2.861572970410884, 3.277313270270915, 4.053326890961483, 4.388891974949197, 5.030578702273187, 5.711517716457019, 6.108947770443638, 6.782618541225109, 7.217794580925693, 7.764452565728892, 8.149685394945403, 8.783659872055481, 9.226018719475681, 9.598644304658496, 9.982061306539794, 10.84862934910401, 11.06221265645909, 11.55729185263686, 12.12636694976215, 12.58309433356499, 13.23222340730621, 13.46359526173336
