| L(s) = 1 | + 2·7-s − 5·11-s + 5·17-s + 5·19-s + 6·23-s − 4·29-s + 10·31-s + 10·37-s − 5·41-s − 4·43-s − 8·47-s − 3·49-s − 10·53-s − 10·61-s − 3·67-s + 5·73-s − 10·77-s + 10·79-s − 83-s + 9·89-s − 10·97-s − 2·101-s + 16·103-s − 3·107-s + 10·109-s + 15·113-s + 10·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.50·11-s + 1.21·17-s + 1.14·19-s + 1.25·23-s − 0.742·29-s + 1.79·31-s + 1.64·37-s − 0.780·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 1.37·53-s − 1.28·61-s − 0.366·67-s + 0.585·73-s − 1.13·77-s + 1.12·79-s − 0.109·83-s + 0.953·89-s − 1.01·97-s − 0.199·101-s + 1.57·103-s − 0.290·107-s + 0.957·109-s + 1.41·113-s + 0.916·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.209612821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.209612821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−7.86531297976007042176183485113, −7.52205592705878887861710802926, −6.51537696642957007869951720667, −5.69092876053100759642022670924, −4.97718672508255631219311350055, −4.69318897016512487664067903161, −3.26544206857742223694831063217, −2.91525535700047228707697545627, −1.73264061228553645833382248076, −0.77773671864837701276369099791,
0.77773671864837701276369099791, 1.73264061228553645833382248076, 2.91525535700047228707697545627, 3.26544206857742223694831063217, 4.69318897016512487664067903161, 4.97718672508255631219311350055, 5.69092876053100759642022670924, 6.51537696642957007869951720667, 7.52205592705878887861710802926, 7.86531297976007042176183485113
