| L(s) = 1 | − 2·5-s + 4·11-s − 4·13-s + 6·17-s − 4·19-s + 4·23-s − 25-s + 14·41-s + 16·43-s − 2·49-s + 14·53-s − 8·55-s − 20·59-s + 8·65-s + 8·67-s + 16·71-s + 20·73-s − 12·85-s + 14·89-s + 8·95-s + 24·97-s − 12·103-s + 24·107-s − 26·109-s − 2·113-s − 8·115-s + 8·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 2.18·41-s + 2.43·43-s − 2/7·49-s + 1.92·53-s − 1.07·55-s − 2.60·59-s + 0.992·65-s + 0.977·67-s + 1.89·71-s + 2.34·73-s − 1.30·85-s + 1.48·89-s + 0.820·95-s + 2.43·97-s − 1.18·103-s + 2.32·107-s − 2.49·109-s − 0.188·113-s − 0.746·115-s + 8/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527678595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527678595\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−9.206150558778119962482385793940, −8.974212960437077931569025427222, −8.240406131633409502091524494819, −7.85928037784726187327192846261, −7.84362018068616659770184084899, −7.23264967566651945313444786332, −7.01064338397204336595553298079, −6.52323617892311366008963575075, −6.03726173065946017773684738263, −5.69967711062826022657953556198, −5.25664457529297197103937870738, −4.67239527632628395405512880236, −4.18913666978169633703951436626, −4.14329189997210722405615489501, −3.31073452331715195145118575503, −3.24449785326798245157571115658, −2.20323720248204202741006409580, −2.17573382884980601700993316084, −0.901533895523881195333624774647, −0.74779677749096740819125089662,
0.74779677749096740819125089662, 0.901533895523881195333624774647, 2.17573382884980601700993316084, 2.20323720248204202741006409580, 3.24449785326798245157571115658, 3.31073452331715195145118575503, 4.14329189997210722405615489501, 4.18913666978169633703951436626, 4.67239527632628395405512880236, 5.25664457529297197103937870738, 5.69967711062826022657953556198, 6.03726173065946017773684738263, 6.52323617892311366008963575075, 7.01064338397204336595553298079, 7.23264967566651945313444786332, 7.84362018068616659770184084899, 7.85928037784726187327192846261, 8.240406131633409502091524494819, 8.974212960437077931569025427222, 9.206150558778119962482385793940
