| L(s) = 1 | + 11-s + 4·13-s − 2·17-s − 4·19-s − 7·23-s + 9·29-s − 2·31-s − 37-s + 8·41-s − 9·43-s + 4·47-s + 6·53-s − 4·59-s − 4·61-s + 9·67-s + 5·71-s + 10·73-s + 15·79-s − 6·83-s + 8·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.45·23-s + 1.67·29-s − 0.359·31-s − 0.164·37-s + 1.24·41-s − 1.37·43-s + 0.583·47-s + 0.824·53-s − 0.520·59-s − 0.512·61-s + 1.09·67-s + 0.593·71-s + 1.17·73-s + 1.68·79-s − 0.658·83-s + 0.847·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501361493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501361493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.28006635456999, −12.56523096193947, −12.36962175677817, −11.73090005200344, −11.30552219542655, −10.81920653251911, −10.31415864754318, −10.03254298550421, −9.205185441907801, −8.949007699031978, −8.290687735018687, −8.092396349353950, −7.435911050370227, −6.643591938154362, −6.404231838008579, −6.028965990804293, −5.310691644205339, −4.722019537340856, −4.134955442912550, −3.771933593093664, −3.168959050031742, −2.304389129092849, −1.989230492124710, −1.128800584046701, −0.4913377034764745,
0.4913377034764745, 1.128800584046701, 1.989230492124710, 2.304389129092849, 3.168959050031742, 3.771933593093664, 4.134955442912550, 4.722019537340856, 5.310691644205339, 6.028965990804293, 6.404231838008579, 6.643591938154362, 7.435911050370227, 8.092396349353950, 8.290687735018687, 8.949007699031978, 9.205185441907801, 10.03254298550421, 10.31415864754318, 10.81920653251911, 11.30552219542655, 11.73090005200344, 12.36962175677817, 12.56523096193947, 13.28006635456999
