| L(s) = 1 | + 48·7-s + 16·31-s + 1.24e3·49-s + 320·73-s − 400·79-s − 448·97-s − 176·103-s − 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 768·217-s + ⋯ |
| L(s) = 1 | + 48/7·7-s + 0.516·31-s + 25.3·49-s + 4.38·73-s − 5.06·79-s − 4.61·97-s − 1.70·103-s − 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 3.53·217-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.26516506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.26516506\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 466 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 510 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 800 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1838 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2912 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3634 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4130 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3168 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 690 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7246 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 1234 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9282 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 100 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 3150 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 6208 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 112 T + p^{2} T^{2} )^{4} \) |
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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
−6.25231970839763618273844285098, −5.72642472179691822187391424798, −5.52911379215343922349552272319, −5.52828290946116994943836104298, −5.17990542197228296123252643727, −5.12128550886938355512405107694, −5.02434590085314595241418258265, −4.87321722919007080288302584421, −4.56767823189433713174181826525, −4.20112616256996248208562055703, −4.15618141018812305370851210340, −4.10926573972510453010077376445, −3.96012697652975078134425815278, −3.52448194525893440379646586105, −2.77580557441455400694131724792, −2.70426411494915098623203389753, −2.68650048211361847914081194775, −2.19143316593271686386545506957, −1.84442371191743618286855355149, −1.80870223675713473604432218662, −1.60310504470730833133508017934, −1.22200762364423175530284673438, −1.08732325680153495725189712415, −1.04004376914442685464947457849, −0.29494889229462058255480816549,
0.29494889229462058255480816549, 1.04004376914442685464947457849, 1.08732325680153495725189712415, 1.22200762364423175530284673438, 1.60310504470730833133508017934, 1.80870223675713473604432218662, 1.84442371191743618286855355149, 2.19143316593271686386545506957, 2.68650048211361847914081194775, 2.70426411494915098623203389753, 2.77580557441455400694131724792, 3.52448194525893440379646586105, 3.96012697652975078134425815278, 4.10926573972510453010077376445, 4.15618141018812305370851210340, 4.20112616256996248208562055703, 4.56767823189433713174181826525, 4.87321722919007080288302584421, 5.02434590085314595241418258265, 5.12128550886938355512405107694, 5.17990542197228296123252643727, 5.52828290946116994943836104298, 5.52911379215343922349552272319, 5.72642472179691822187391424798, 6.25231970839763618273844285098
