| L(s) = 1 | − 3-s − 2·4-s + 2·12-s − 13-s + 3·16-s + 3·19-s + 25-s + 27-s + 39-s + 43-s − 3·48-s + 2·52-s − 3·57-s + 61-s − 4·64-s + 3·73-s − 75-s − 6·76-s + 2·79-s − 81-s + 3·97-s − 2·100-s − 103-s − 2·108-s + 3·109-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | − 3-s − 2·4-s + 2·12-s − 13-s + 3·16-s + 3·19-s + 25-s + 27-s + 39-s + 43-s − 3·48-s + 2·52-s − 3·57-s + 61-s − 4·64-s + 3·73-s − 75-s − 6·76-s + 2·79-s − 81-s + 3·97-s − 2·100-s − 103-s − 2·108-s + 3·109-s + 121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5303760414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5303760414\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.443509825291709453415471530647, −9.377586759013978429858816401509, −8.942857247351900771531994795596, −8.473562133858525512805613385050, −8.006726202537607326669702055287, −7.76089650629337650587065927261, −7.13088013422529407678909832241, −7.07993286125100485321485777199, −6.09768905813057962388267373373, −5.97975760862955154228519980021, −5.35251650774687348462217162307, −5.08270375959676737926421746361, −4.87708361920055056771955686231, −4.61213433406141015990881293163, −3.64692694192492682310858932750, −3.56822761450352521665703062324, −2.98584543240951196608292332664, −2.25026882348010017562896414617, −0.955850460410881885712347424039, −0.845312294892577637528653377864,
0.845312294892577637528653377864, 0.955850460410881885712347424039, 2.25026882348010017562896414617, 2.98584543240951196608292332664, 3.56822761450352521665703062324, 3.64692694192492682310858932750, 4.61213433406141015990881293163, 4.87708361920055056771955686231, 5.08270375959676737926421746361, 5.35251650774687348462217162307, 5.97975760862955154228519980021, 6.09768905813057962388267373373, 7.07993286125100485321485777199, 7.13088013422529407678909832241, 7.76089650629337650587065927261, 8.006726202537607326669702055287, 8.473562133858525512805613385050, 8.942857247351900771531994795596, 9.377586759013978429858816401509, 9.443509825291709453415471530647
