| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 4·11-s + 2·12-s + 16-s + 4·27-s + 8·33-s + 3·36-s + 4·37-s + 4·44-s − 16·47-s + 2·48-s + 2·49-s + 4·53-s + 8·59-s + 64-s + 8·67-s + 16·71-s + 5·81-s − 12·89-s − 4·97-s + 12·99-s + 16·103-s + 4·108-s + 8·111-s + 12·113-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 1.20·11-s + 0.577·12-s + 1/4·16-s + 0.769·27-s + 1.39·33-s + 1/2·36-s + 0.657·37-s + 0.603·44-s − 2.33·47-s + 0.288·48-s + 2/7·49-s + 0.549·53-s + 1.04·59-s + 1/8·64-s + 0.977·67-s + 1.89·71-s + 5/9·81-s − 1.27·89-s − 0.406·97-s + 1.20·99-s + 1.57·103-s + 0.384·108-s + 0.759·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.036083569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.036083569\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.64880829766810418699486714711, −7.10290424063309812705086492771, −6.90002773314140894650694292406, −6.44697342607198049762532503663, −6.11427480647466147063394540009, −5.47908178907939397008520804639, −5.00149666304429360153128700827, −4.44953894616674393482931521847, −4.00406209704214437790181822384, −3.53156936040376420905508137127, −3.22330894997349096623134014751, −2.53728880200607898832561922674, −2.08606865763887064989603594611, −1.53339721829519230746175608522, −0.834678242396562007160680067169,
0.834678242396562007160680067169, 1.53339721829519230746175608522, 2.08606865763887064989603594611, 2.53728880200607898832561922674, 3.22330894997349096623134014751, 3.53156936040376420905508137127, 4.00406209704214437790181822384, 4.44953894616674393482931521847, 5.00149666304429360153128700827, 5.47908178907939397008520804639, 6.11427480647466147063394540009, 6.44697342607198049762532503663, 6.90002773314140894650694292406, 7.10290424063309812705086492771, 7.64880829766810418699486714711
