| L(s) = 1 | + 2·5-s + 7-s − 6·11-s + 4·17-s − 19-s − 4·23-s − 25-s + 2·29-s − 2·31-s + 2·35-s + 10·41-s − 4·43-s − 4·47-s + 49-s + 14·53-s − 12·55-s − 2·61-s + 12·67-s − 2·73-s − 6·77-s + 10·79-s + 6·83-s + 8·85-s + 10·89-s − 2·95-s − 2·97-s + 6·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.80·11-s + 0.970·17-s − 0.229·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s + 0.338·35-s + 1.56·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.92·53-s − 1.61·55-s − 0.256·61-s + 1.46·67-s − 0.234·73-s − 0.683·77-s + 1.12·79-s + 0.658·83-s + 0.867·85-s + 1.05·89-s − 0.205·95-s − 0.203·97-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174555550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174555550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.954215306167202474143682901386, −7.06376907928095257268700852721, −6.15443643126058538256986707565, −5.54392408701394582278901915195, −5.18464563248056315137994876658, −4.28002375858814927962441320747, −3.32095948386804286428434627535, −2.43075790737746048950037875853, −1.93609286510072722993651107570, −0.68589763845831653800569624817,
0.68589763845831653800569624817, 1.93609286510072722993651107570, 2.43075790737746048950037875853, 3.32095948386804286428434627535, 4.28002375858814927962441320747, 5.18464563248056315137994876658, 5.54392408701394582278901915195, 6.15443643126058538256986707565, 7.06376907928095257268700852721, 7.954215306167202474143682901386
