| L(s) = 1 | − 5-s − 2·11-s − 5·13-s + 4·17-s − 2·19-s + 23-s + 25-s + 3·29-s + 7·31-s − 2·37-s + 9·41-s − 4·43-s + 9·47-s − 7·49-s + 6·53-s + 2·55-s + 2·61-s + 5·65-s − 2·67-s + 71-s + 73-s − 14·79-s − 4·85-s − 16·89-s + 2·95-s − 4·97-s − 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 1.38·13-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.557·29-s + 1.25·31-s − 0.328·37-s + 1.40·41-s − 0.609·43-s + 1.31·47-s − 49-s + 0.824·53-s + 0.269·55-s + 0.256·61-s + 0.620·65-s − 0.244·67-s + 0.118·71-s + 0.117·73-s − 1.57·79-s − 0.433·85-s − 1.69·89-s + 0.205·95-s − 0.406·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.44562594653426746837768678727, −6.96384478949261151934855166781, −6.04308041519196586441540983421, −5.29796325657690428899962089629, −4.67075715269829331721312412913, −3.97482952818300810701923022657, −2.91103419166611649736344659614, −2.46110585384113311166655930549, −1.14044450654319127021785432329, 0,
1.14044450654319127021785432329, 2.46110585384113311166655930549, 2.91103419166611649736344659614, 3.97482952818300810701923022657, 4.67075715269829331721312412913, 5.29796325657690428899962089629, 6.04308041519196586441540983421, 6.96384478949261151934855166781, 7.44562594653426746837768678727
