| L(s) = 1 | − 4·4-s + 12·16-s + 68·25-s − 128·37-s + 80·43-s − 32·64-s − 192·67-s − 592·79-s − 272·100-s − 344·109-s + 468·121-s + 127-s + 131-s + 137-s + 139-s + 512·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 352·169-s − 320·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 2.71·25-s − 3.45·37-s + 1.86·43-s − 1/2·64-s − 2.86·67-s − 7.49·79-s − 2.71·100-s − 3.15·109-s + 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.45·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.08·169-s − 1.86·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.552794170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552794170\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 234 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 562 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 294 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 624 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1918 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 4018 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3360 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6946 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4250 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 3792 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 148 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 7378 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 4606 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 4944 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.10278913808860247598617952721, −6.94270571434340631402145303890, −6.90820627297090528649355359121, −6.25044464049214110665851980855, −5.91528427975366815224190816276, −5.90900750799775535169453624493, −5.87345101926073778281718802119, −5.35415218728470933272832447011, −5.00481483976617660077083118511, −4.99627272396817247424165704985, −4.89358714918937044949945249918, −4.33275798585194788503316457314, −4.13768526888750765063140877640, −4.11187377432161820979494012702, −3.84254587389051184293736457857, −3.15020572701284162199747998256, −3.05492151474207532749374526011, −2.93147448527198877690404276892, −2.82402918858490982687021733281, −2.12389168712459658678415954729, −1.71212627479738322386044140685, −1.38194832148221380729769100428, −1.28140172616262453753795129192, −0.56193555964805536961257468428, −0.27229488788256116857157637041,
0.27229488788256116857157637041, 0.56193555964805536961257468428, 1.28140172616262453753795129192, 1.38194832148221380729769100428, 1.71212627479738322386044140685, 2.12389168712459658678415954729, 2.82402918858490982687021733281, 2.93147448527198877690404276892, 3.05492151474207532749374526011, 3.15020572701284162199747998256, 3.84254587389051184293736457857, 4.11187377432161820979494012702, 4.13768526888750765063140877640, 4.33275798585194788503316457314, 4.89358714918937044949945249918, 4.99627272396817247424165704985, 5.00481483976617660077083118511, 5.35415218728470933272832447011, 5.87345101926073778281718802119, 5.90900750799775535169453624493, 5.91528427975366815224190816276, 6.25044464049214110665851980855, 6.90820627297090528649355359121, 6.94270571434340631402145303890, 7.10278913808860247598617952721
