| L(s) = 1 | + 3-s − 5-s − 3·7-s − 9-s + 2·11-s − 4·13-s − 15-s + 2·17-s + 19-s − 3·21-s + 6·23-s − 5·25-s + 9·29-s + 4·31-s + 2·33-s + 3·35-s − 4·37-s − 4·39-s − 11·41-s − 16·43-s + 45-s − 2·47-s − 3·49-s + 2·51-s − 3·53-s − 2·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s − 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.654·21-s + 1.25·23-s − 25-s + 1.67·29-s + 0.718·31-s + 0.348·33-s + 0.507·35-s − 0.657·37-s − 0.640·39-s − 1.71·41-s − 2.43·43-s + 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.280·51-s − 0.412·53-s − 0.269·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64384576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64384576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 156 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.80738419845090117104302167113, −7.12692911406945995050820689645, −7.05030090848393896217397584038, −6.60828109491768932842493557701, −6.29594045574232321259216244909, −6.28810607645959460613950538951, −5.42376584715145932476982736357, −5.04067500901734614046889511784, −4.94925579545111104286341970821, −4.61340097443610130069245139158, −3.77774943212923237161304977074, −3.69443303508238493960751834237, −3.22775350615148981000207304643, −3.04903918428058675701340130318, −2.55387407158307993426290374595, −2.17833500422004616671342476731, −1.41633302698794609502255218889, −1.09513987731390760121517679020, 0, 0,
1.09513987731390760121517679020, 1.41633302698794609502255218889, 2.17833500422004616671342476731, 2.55387407158307993426290374595, 3.04903918428058675701340130318, 3.22775350615148981000207304643, 3.69443303508238493960751834237, 3.77774943212923237161304977074, 4.61340097443610130069245139158, 4.94925579545111104286341970821, 5.04067500901734614046889511784, 5.42376584715145932476982736357, 6.28810607645959460613950538951, 6.29594045574232321259216244909, 6.60828109491768932842493557701, 7.05030090848393896217397584038, 7.12692911406945995050820689645, 7.80738419845090117104302167113
