| L(s) = 1 | + 4·7-s − 11-s − 6·13-s − 6·17-s − 4·19-s − 4·23-s + 2·29-s − 8·31-s + 10·37-s − 10·41-s − 4·47-s + 9·49-s − 10·53-s − 4·59-s − 2·61-s − 8·67-s + 14·73-s − 4·77-s + 16·79-s + 8·83-s + 6·89-s − 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.301·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s − 1.43·31-s + 1.64·37-s − 1.56·41-s − 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.977·67-s + 1.63·73-s − 0.455·77-s + 1.80·79-s + 0.878·83-s + 0.635·89-s − 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.154496600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154496600\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.93719500517563, −14.29795791633874, −13.92887039558798, −13.16042398025185, −12.76158068889969, −12.12019357967023, −11.64699512281116, −11.11436601366636, −10.69407827424018, −10.14586559947530, −9.418742314029574, −8.976176561292709, −8.276790932253892, −7.783348122340537, −7.490959488468550, −6.588041266368004, −6.218220542483476, −5.143659821510484, −4.910863805935629, −4.437072559458446, −3.715139352843885, −2.653556615434410, −2.080776909293105, −1.712876515326923, −0.3660211538825820,
0.3660211538825820, 1.712876515326923, 2.080776909293105, 2.653556615434410, 3.715139352843885, 4.437072559458446, 4.910863805935629, 5.143659821510484, 6.218220542483476, 6.588041266368004, 7.490959488468550, 7.783348122340537, 8.276790932253892, 8.976176561292709, 9.418742314029574, 10.14586559947530, 10.69407827424018, 11.11436601366636, 11.64699512281116, 12.12019357967023, 12.76158068889969, 13.16042398025185, 13.92887039558798, 14.29795791633874, 14.93719500517563
