| L(s) = 1 | + 2·7-s − 14·13-s + 14·19-s − 34·31-s − 32·37-s + 110·43-s − 95·49-s + 130·61-s + 98·67-s − 176·73-s + 80·79-s − 28·91-s + 82·97-s + 164·103-s − 98·109-s + 224·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + ⋯ |
| L(s) = 1 | + 2/7·7-s − 1.07·13-s + 0.736·19-s − 1.09·31-s − 0.864·37-s + 2.55·43-s − 1.93·49-s + 2.13·61-s + 1.46·67-s − 2.41·73-s + 1.01·79-s − 0.307·91-s + 0.845·97-s + 1.59·103-s − 0.899·109-s + 1.85·121-s + 0.00787·127-s + 0.00763·131-s + 4/19·133-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 − 1.13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.792535231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.792535231\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 176 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 800 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 55 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2240 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1582 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3920 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 88 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10864 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 41 T + p^{2} T^{2} )^{2} \) |
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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
−8.388422888110557953712572865116, −8.351363166840256717592096471096, −7.74740623694823761768945135463, −7.37685113474329836739302329975, −7.12712416863195756376831886078, −6.96942303960624916577626608221, −6.20127897586260299341193189718, −5.99418614323939352751151892104, −5.48366059946885323058877827286, −5.20033029440192045590276830542, −4.74912666252907551970200623628, −4.49245980780366667095918110311, −3.80276228364354502215896742288, −3.60327175062953656693463852213, −2.97425236934304372344007653325, −2.59913710516308435745800572441, −1.97933735300807524860511441836, −1.72217625216579342263219356064, −0.849482098301961590804494222537, −0.44027002925918458592135016599,
0.44027002925918458592135016599, 0.849482098301961590804494222537, 1.72217625216579342263219356064, 1.97933735300807524860511441836, 2.59913710516308435745800572441, 2.97425236934304372344007653325, 3.60327175062953656693463852213, 3.80276228364354502215896742288, 4.49245980780366667095918110311, 4.74912666252907551970200623628, 5.20033029440192045590276830542, 5.48366059946885323058877827286, 5.99418614323939352751151892104, 6.20127897586260299341193189718, 6.96942303960624916577626608221, 7.12712416863195756376831886078, 7.37685113474329836739302329975, 7.74740623694823761768945135463, 8.351363166840256717592096471096, 8.388422888110557953712572865116
