| L(s) = 1 | − 12·5-s + 18·11-s − 24·17-s − 36·19-s + 18·23-s + 71·25-s − 36·29-s + 72·31-s − 10·37-s − 76·43-s − 48·47-s + 18·53-s − 216·55-s − 12·59-s + 36·61-s − 26·67-s − 36·71-s + 72·73-s − 2·79-s + 288·85-s + 432·95-s − 324·101-s + 144·103-s − 126·107-s + 134·109-s + 252·113-s − 216·115-s + ⋯ |
| L(s) = 1 | − 2.39·5-s + 1.63·11-s − 1.41·17-s − 1.89·19-s + 0.782·23-s + 2.83·25-s − 1.24·29-s + 2.32·31-s − 0.270·37-s − 1.76·43-s − 1.02·47-s + 0.339·53-s − 3.92·55-s − 0.203·59-s + 0.590·61-s − 0.388·67-s − 0.507·71-s + 0.986·73-s − 0.0253·79-s + 3.38·85-s + 4.54·95-s − 3.20·101-s + 1.39·103-s − 1.17·107-s + 1.22·109-s + 2.23·113-s − 1.87·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3567713730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3567713730\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T + 203 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 94 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T + 481 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T + 793 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 18 T - 205 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 - 13 T + p^{2} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 10 T - 1269 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 48 T + 2977 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18 T - 2485 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 3529 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 36 T + 4153 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 26 T - 3813 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 72 T + 7057 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 6237 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12578 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 8830 T^{2} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.220001213767988913881774912242, −8.729074709785002107294895842075, −8.352031472753852627172532131083, −8.290463860767702650635654149160, −7.942299274475535682233556841968, −7.08823181594114565832880818980, −6.98985513780960844551951054590, −6.75610700545566108851327324344, −6.25014630580449280442715783303, −5.84173073415617202132412683905, −4.88252310489212642173445326728, −4.49158169800590299744242319412, −4.48126399582787089237848580191, −3.84610315383280246060782498369, −3.58682431071813444277429347896, −3.11023986358569994377124472842, −2.34347056869838676008781253905, −1.73613578464718103571065393631, −0.950374144930349357113845156824, −0.19316223854646516509323512312,
0.19316223854646516509323512312, 0.950374144930349357113845156824, 1.73613578464718103571065393631, 2.34347056869838676008781253905, 3.11023986358569994377124472842, 3.58682431071813444277429347896, 3.84610315383280246060782498369, 4.48126399582787089237848580191, 4.49158169800590299744242319412, 4.88252310489212642173445326728, 5.84173073415617202132412683905, 6.25014630580449280442715783303, 6.75610700545566108851327324344, 6.98985513780960844551951054590, 7.08823181594114565832880818980, 7.942299274475535682233556841968, 8.290463860767702650635654149160, 8.352031472753852627172532131083, 8.729074709785002107294895842075, 9.220001213767988913881774912242
