L(s) = 1 | + 4·4-s − 8·7-s − 2·9-s + 12·16-s − 2·25-s − 32·28-s + 16·31-s − 8·36-s − 32·41-s − 16·47-s + 16·49-s + 16·63-s + 32·64-s + 8·71-s + 40·73-s + 16·79-s + 3·81-s + 8·89-s − 24·97-s − 8·100-s + 8·103-s − 96·112-s + 48·113-s + 20·121-s + 64·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 2·4-s − 3.02·7-s − 2/3·9-s + 3·16-s − 2/5·25-s − 6.04·28-s + 2.87·31-s − 4/3·36-s − 4.99·41-s − 2.33·47-s + 16/7·49-s + 2.01·63-s + 4·64-s + 0.949·71-s + 4.68·73-s + 1.80·79-s + 1/3·81-s + 0.847·89-s − 2.43·97-s − 4/5·100-s + 0.788·103-s − 9.07·112-s + 4.51·113-s + 1.81·121-s + 5.74·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5809895731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5809895731\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 7 | $D_{4}$ | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 414 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1590 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11286 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 244 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.63936620394605906386408731069, −6.53111915405996632815655851254, −6.47455221693267939520418297351, −6.23359293566417105726757317582, −6.00332504028517107001877758701, −5.96631532074561043772090171438, −5.39138410246780515870443764842, −5.17790722183384553129135132253, −5.15391830942896605490556932038, −4.80152857425384719392127801309, −4.71243635995629284807116286703, −4.14967293699980985124666823655, −3.67072693385860023285196093991, −3.63788087916255792478830394200, −3.28152628471359816703317899874, −3.25293868261969331399702894511, −3.21610395328258209668244637367, −2.93268023043620584814641290223, −2.45252530245709299287790354234, −2.23592910235349755036143940126, −1.96400827553388322132464803786, −1.79837612912412369149625355229, −1.10285649663368728261398752260, −0.871538998233061268934599877208, −0.14547185425451951508320903358,
0.14547185425451951508320903358, 0.871538998233061268934599877208, 1.10285649663368728261398752260, 1.79837612912412369149625355229, 1.96400827553388322132464803786, 2.23592910235349755036143940126, 2.45252530245709299287790354234, 2.93268023043620584814641290223, 3.21610395328258209668244637367, 3.25293868261969331399702894511, 3.28152628471359816703317899874, 3.63788087916255792478830394200, 3.67072693385860023285196093991, 4.14967293699980985124666823655, 4.71243635995629284807116286703, 4.80152857425384719392127801309, 5.15391830942896605490556932038, 5.17790722183384553129135132253, 5.39138410246780515870443764842, 5.96631532074561043772090171438, 6.00332504028517107001877758701, 6.23359293566417105726757317582, 6.47455221693267939520418297351, 6.53111915405996632815655851254, 6.63936620394605906386408731069