L(s) = 1 | + 2·9-s − 24·19-s − 30·25-s + 136·31-s + 68·49-s + 296·61-s − 312·79-s − 77·81-s + 296·109-s + 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 388·169-s − 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 2/9·9-s − 1.26·19-s − 6/5·25-s + 4.38·31-s + 1.38·49-s + 4.85·61-s − 3.94·79-s − 0.950·81-s + 2.71·109-s + 2.67·121-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 + 2.29·169-s − 0.280·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.518675674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.518675674\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 p T^{2} + p^{4} T^{4} \) |
good | 7 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 402 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1038 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3042 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4398 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2718 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 514 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7522 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15522 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17794 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−11.30211096781844578266825975358, −10.55323521472925389904880559745, −10.12245583978878001683934168376, −10.05556224265148204549474634162, −10.04693880366742053660088226793, −9.694539967310661605320113102478, −8.880679047898305080751982526464, −8.850276241935608498957155139665, −8.347297233587131691794432032574, −8.289150755434274122346210321874, −7.979825964955332226259643904789, −7.26047177347605434689188772540, −7.18236054184151654360121377292, −6.70381367980156212657825159348, −6.37790850033422000708432325114, −5.94137585119197281268068358720, −5.73247912943135476439684726602, −5.12683592095536953050554062765, −4.62424247316482503301892295422, −4.17989063800333591392503030492, −4.08223328678377031679040300796, −3.23021354165468243900978440290, −2.58186362428924905480915729866, −2.17131723727890984097766088136, −0.940269221245899440138908656911,
0.940269221245899440138908656911, 2.17131723727890984097766088136, 2.58186362428924905480915729866, 3.23021354165468243900978440290, 4.08223328678377031679040300796, 4.17989063800333591392503030492, 4.62424247316482503301892295422, 5.12683592095536953050554062765, 5.73247912943135476439684726602, 5.94137585119197281268068358720, 6.37790850033422000708432325114, 6.70381367980156212657825159348, 7.18236054184151654360121377292, 7.26047177347605434689188772540, 7.979825964955332226259643904789, 8.289150755434274122346210321874, 8.347297233587131691794432032574, 8.850276241935608498957155139665, 8.880679047898305080751982526464, 9.694539967310661605320113102478, 10.04693880366742053660088226793, 10.05556224265148204549474634162, 10.12245583978878001683934168376, 10.55323521472925389904880559745, 11.30211096781844578266825975358