| L(s) = 1 | + 5-s − 3·7-s − 3·11-s + 4·17-s + 6·19-s − 6·23-s − 4·25-s + 2·29-s − 9·31-s − 3·35-s + 2·37-s − 10·41-s + 6·43-s − 6·47-s + 2·49-s − 13·53-s − 3·55-s − 12·59-s − 8·61-s + 6·67-s − 12·71-s + 9·73-s + 9·77-s − 3·83-s + 4·85-s + 14·89-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 0.904·11-s + 0.970·17-s + 1.37·19-s − 1.25·23-s − 4/5·25-s + 0.371·29-s − 1.61·31-s − 0.507·35-s + 0.328·37-s − 1.56·41-s + 0.914·43-s − 0.875·47-s + 2/7·49-s − 1.78·53-s − 0.404·55-s − 1.56·59-s − 1.02·61-s + 0.733·67-s − 1.42·71-s + 1.05·73-s + 1.02·77-s − 0.329·83-s + 0.433·85-s + 1.48·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−9.175179377906198062244226956130, −7.929463326432325863526345920674, −7.50102058506384286932539601186, −6.35310757105558722450465771380, −5.76019206321813718877416408605, −4.97608450620626805785013090883, −3.60003880842544419710753739473, −2.99069988080223923000922886378, −1.71409356086345270227455181385, 0,
1.71409356086345270227455181385, 2.99069988080223923000922886378, 3.60003880842544419710753739473, 4.97608450620626805785013090883, 5.76019206321813718877416408605, 6.35310757105558722450465771380, 7.50102058506384286932539601186, 7.929463326432325863526345920674, 9.175179377906198062244226956130
