| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·15-s + 8·23-s + 3·25-s + 4·27-s − 8·31-s + 12·37-s − 6·45-s − 8·47-s + 2·49-s + 4·53-s − 8·67-s + 16·69-s + 6·75-s + 5·81-s + 20·89-s − 16·93-s + 4·97-s + 16·103-s + 24·111-s + 4·113-s − 16·115-s − 11·121-s − 4·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.03·15-s + 1.66·23-s + 3/5·25-s + 0.769·27-s − 1.43·31-s + 1.97·37-s − 0.894·45-s − 1.16·47-s + 2/7·49-s + 0.549·53-s − 0.977·67-s + 1.92·69-s + 0.692·75-s + 5/9·81-s + 2.11·89-s − 1.65·93-s + 0.406·97-s + 1.57·103-s + 2.27·111-s + 0.376·113-s − 1.49·115-s − 121-s − 0.357·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.885281286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.885281286\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.71528101906085669647375040654, −7.63713415853611256613927117267, −7.08519898122851311918320683289, −6.75481797909217947269819574815, −6.20106393406836656007681791180, −5.67195016762935757324905905330, −4.96362174580825642245676512711, −4.74154141095460092284244168168, −4.14640322325407711686489960274, −3.68281757322116344142767058321, −3.26540771865108010376968618927, −2.81013649115573541339950188327, −2.24340920942870394733528685226, −1.48435987976982719209824395532, −0.68859665404839937867782964493,
0.68859665404839937867782964493, 1.48435987976982719209824395532, 2.24340920942870394733528685226, 2.81013649115573541339950188327, 3.26540771865108010376968618927, 3.68281757322116344142767058321, 4.14640322325407711686489960274, 4.74154141095460092284244168168, 4.96362174580825642245676512711, 5.67195016762935757324905905330, 6.20106393406836656007681791180, 6.75481797909217947269819574815, 7.08519898122851311918320683289, 7.63713415853611256613927117267, 7.71528101906085669647375040654
