| L(s) = 1 | + 2·3-s + 9-s − 11-s − 3·13-s + 4·17-s + 19-s + 3·23-s − 4·27-s + 5·29-s + 3·31-s − 2·33-s + 12·37-s − 6·39-s + 8·41-s − 5·43-s − 8·47-s − 7·49-s + 8·51-s + 10·53-s + 2·57-s − 8·59-s + 10·61-s + 14·67-s + 6·69-s + 5·71-s + 4·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.970·17-s + 0.229·19-s + 0.625·23-s − 0.769·27-s + 0.928·29-s + 0.538·31-s − 0.348·33-s + 1.97·37-s − 0.960·39-s + 1.24·41-s − 0.762·43-s − 1.16·47-s − 49-s + 1.12·51-s + 1.37·53-s + 0.264·57-s − 1.04·59-s + 1.28·61-s + 1.71·67-s + 0.722·69-s + 0.593·71-s + 0.468·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879525306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879525306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.143678081403822547014039537933, −7.924150992289578066135677247704, −7.10641013336201846400301506184, −6.23740892709688388520338017610, −5.29978594590640349394342386550, −4.58425330888590593193754640112, −3.57251191371041815374631189621, −2.85753582074253957031265532877, −2.25174693314083683559479568036, −0.909335239692207586223919408186,
0.909335239692207586223919408186, 2.25174693314083683559479568036, 2.85753582074253957031265532877, 3.57251191371041815374631189621, 4.58425330888590593193754640112, 5.29978594590640349394342386550, 6.23740892709688388520338017610, 7.10641013336201846400301506184, 7.924150992289578066135677247704, 8.143678081403822547014039537933
