| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 2·11-s + 2·13-s + 15-s − 4·17-s + 6·19-s + 2·21-s + 6·23-s + 25-s − 27-s + 10·29-s − 8·31-s − 2·33-s + 2·35-s − 2·37-s − 2·39-s + 2·41-s + 43-s − 45-s − 2·47-s − 3·49-s + 4·51-s − 10·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.560·51-s − 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.723692521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.723692521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.76600491081445, −14.15920744902071, −13.74040797401908, −13.02307583291983, −12.68586386352566, −12.18350392548023, −11.51565085031852, −11.13582986300305, −10.76557886924323, −9.978574736967670, −9.393253540103094, −9.108528819830355, −8.338249893096395, −7.791690670415072, −6.945323451060227, −6.676858764364722, −6.273350668883628, −5.257219696134904, −5.041547335745796, −4.164203432066624, −3.539777221299978, −3.098570335000673, −2.152547463041065, −1.141615309307555, −0.5774708248849159,
0.5774708248849159, 1.141615309307555, 2.152547463041065, 3.098570335000673, 3.539777221299978, 4.164203432066624, 5.041547335745796, 5.257219696134904, 6.273350668883628, 6.676858764364722, 6.945323451060227, 7.791690670415072, 8.338249893096395, 9.108528819830355, 9.393253540103094, 9.978574736967670, 10.76557886924323, 11.13582986300305, 11.51565085031852, 12.18350392548023, 12.68586386352566, 13.02307583291983, 13.74040797401908, 14.15920744902071, 14.76600491081445
