| L(s) = 1 | + 16·7-s − 14·9-s + 32·23-s − 48·29-s − 188·41-s − 224·43-s + 304·47-s + 52·49-s + 152·61-s − 224·63-s + 128·67-s − 15·81-s − 160·83-s + 276·89-s + 296·101-s − 16·103-s − 40·109-s + 334·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 512·161-s + 163-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 1.55·9-s + 1.39·23-s − 1.65·29-s − 4.58·41-s − 5.20·43-s + 6.46·47-s + 1.06·49-s + 2.49·61-s − 3.55·63-s + 1.91·67-s − 0.185·81-s − 1.92·83-s + 3.10·89-s + 2.93·101-s − 0.155·103-s − 0.366·109-s + 2.76·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 3.18·161-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.529027312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.529027312\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 8 T + 10 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 334 T^{2} + 54355 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 21670 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 302 T^{2} + 119443 T^{4} + 302 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 623 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 16 T + 1078 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 242 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2540 T^{2} + 3054438 T^{4} - 2540 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 p T^{2} + 4911334 T^{4} - 76 p^{5} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 94 T + 4867 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 112 T + 6658 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 34042918 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12004 T^{2} + 59537830 T^{4} - 12004 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 76 T + 4486 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 64 T + 5151 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 5924 T^{2} + 39727110 T^{4} - 5924 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 2737 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2572 T^{2} + 28232358 T^{4} - 2572 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 80 T + 12903 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 138 T + 14267 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8764 T^{2} - 9025914 T^{4} - 8764 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.46757038370895481986614694670, −6.26232813081026563276470399387, −6.00236010440715616972958853684, −5.84088105207312753429471493412, −5.49481060632694559189554879712, −5.29248823813767672048423291262, −5.19750815393829078872085334972, −5.08017402946005842522621519417, −4.91569659812655017550456992244, −4.62705723783074686399084331793, −4.39037382756987208464444657455, −4.02982300943261731425576149641, −3.80402712235514036971034684608, −3.40069099331744428237506765274, −3.34468542463338775270272500456, −3.15305760622963061261931459134, −2.94801180495317193005733409140, −2.24319752386325128396506865962, −2.08236780554206824658810349483, −2.07136631397024277039220035622, −1.64638106073322158357365296345, −1.58205285592552828385690218290, −0.76483157755378336358698947677, −0.73102016231716119304200985760, −0.33535128237142475297195612118,
0.33535128237142475297195612118, 0.73102016231716119304200985760, 0.76483157755378336358698947677, 1.58205285592552828385690218290, 1.64638106073322158357365296345, 2.07136631397024277039220035622, 2.08236780554206824658810349483, 2.24319752386325128396506865962, 2.94801180495317193005733409140, 3.15305760622963061261931459134, 3.34468542463338775270272500456, 3.40069099331744428237506765274, 3.80402712235514036971034684608, 4.02982300943261731425576149641, 4.39037382756987208464444657455, 4.62705723783074686399084331793, 4.91569659812655017550456992244, 5.08017402946005842522621519417, 5.19750815393829078872085334972, 5.29248823813767672048423291262, 5.49481060632694559189554879712, 5.84088105207312753429471493412, 6.00236010440715616972958853684, 6.26232813081026563276470399387, 6.46757038370895481986614694670
