| L(s) = 1 | + 3-s − 3.46·5-s + 7-s + 9-s − 1.46·11-s − 2·13-s − 3.46·15-s + 0.535·17-s + 6.92·19-s + 21-s + 1.46·23-s + 6.99·25-s + 27-s + 4.92·29-s + 10.9·31-s − 1.46·33-s − 3.46·35-s + 2·37-s − 2·39-s + 11.4·41-s − 8·43-s − 3.46·45-s − 10.9·47-s + 49-s + 0.535·51-s + 2·53-s + 5.07·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54·5-s + 0.377·7-s + 0.333·9-s − 0.441·11-s − 0.554·13-s − 0.894·15-s + 0.129·17-s + 1.58·19-s + 0.218·21-s + 0.305·23-s + 1.39·25-s + 0.192·27-s + 0.915·29-s + 1.96·31-s − 0.254·33-s − 0.585·35-s + 0.328·37-s − 0.320·39-s + 1.79·41-s − 1.21·43-s − 0.516·45-s − 1.59·47-s + 0.142·49-s + 0.0750·51-s + 0.274·53-s + 0.683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570467728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570467728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−9.607696103732311323727337069700, −8.555913734903393850129610394425, −7.908011629731965303335958076443, −7.52561453372793412145057707719, −6.56423374657304839964672267732, −5.09605580276489345368804404109, −4.48575633050017145540742464761, −3.43016756322814507023763083085, −2.67646990808512568486281656535, −0.908602359008531690662611860356,
0.908602359008531690662611860356, 2.67646990808512568486281656535, 3.43016756322814507023763083085, 4.48575633050017145540742464761, 5.09605580276489345368804404109, 6.56423374657304839964672267732, 7.52561453372793412145057707719, 7.908011629731965303335958076443, 8.555913734903393850129610394425, 9.607696103732311323727337069700
