| L(s) = 1 | − 2·3-s + 2·5-s − 9-s − 4·11-s + 4·13-s − 4·15-s − 4·17-s − 14·23-s + 3·25-s + 6·27-s − 10·29-s − 4·31-s + 8·33-s − 8·39-s + 6·41-s + 6·43-s − 2·45-s + 12·47-s + 8·51-s − 8·55-s − 8·59-s + 2·61-s + 8·65-s − 6·67-s + 28·69-s − 24·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 1.03·15-s − 0.970·17-s − 2.91·23-s + 3/5·25-s + 1.15·27-s − 1.85·29-s − 0.718·31-s + 1.39·33-s − 1.28·39-s + 0.937·41-s + 0.914·43-s − 0.298·45-s + 1.75·47-s + 1.12·51-s − 1.07·55-s − 1.04·59-s + 0.256·61-s + 0.992·65-s − 0.733·67-s + 3.37·69-s − 2.84·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14 T + 93 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 91 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.840315981099739602337157609235, −8.834211035385820467997563880257, −8.220199045066586983224569626822, −7.77154923491413003868654168761, −7.25426298782425234681427305257, −7.20311365171144219768442050524, −6.14309172842128982033781547984, −5.99566180364328252561312775060, −5.92658564617297668241953228398, −5.67926474376942607471662334107, −5.09793864942923372879666655740, −4.56668904680839968782865697046, −3.92916244059385306929159040555, −3.86269375915728988627128897120, −2.72846026642977691943098186241, −2.60316358748980387249469785974, −1.87334374716229158862605822539, −1.37308574571468160271085307486, 0, 0,
1.37308574571468160271085307486, 1.87334374716229158862605822539, 2.60316358748980387249469785974, 2.72846026642977691943098186241, 3.86269375915728988627128897120, 3.92916244059385306929159040555, 4.56668904680839968782865697046, 5.09793864942923372879666655740, 5.67926474376942607471662334107, 5.92658564617297668241953228398, 5.99566180364328252561312775060, 6.14309172842128982033781547984, 7.20311365171144219768442050524, 7.25426298782425234681427305257, 7.77154923491413003868654168761, 8.220199045066586983224569626822, 8.834211035385820467997563880257, 8.840315981099739602337157609235
