| L(s) = 1 | − 30·9-s − 48·29-s − 252·41-s − 172·49-s + 104·61-s + 513·81-s − 396·89-s + 600·101-s + 296·109-s + 190·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 164·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.33·9-s − 1.65·29-s − 6.14·41-s − 3.51·49-s + 1.70·61-s + 19/3·81-s − 4.44·89-s + 5.94·101-s + 2.71·109-s + 1.57·121-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 + 0.00613·163-s + 0.00598·167-s + 0.970·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3875605349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3875605349\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 5 p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 569 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 215 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 970 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 950 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 63 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4226 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5294 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3890 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 7895 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9314 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10369 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 4982 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 4031 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 99 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 862 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.043208572205063538089282139160, −7.87713074503043243135403273829, −7.51475251189716923085089632017, −7.13525026459992030418035652186, −6.74195158675404802760331752652, −6.69358011579204881157978676911, −6.60014490214433419660108498344, −5.99319272472723371802268190812, −5.90666910886784429580700605242, −5.59584836624783009560434669158, −5.51594457101220937413974190763, −5.04806526204044684207028691682, −4.90059560415257564000565627835, −4.79801644608287067850378744332, −4.24767152600730571291701615058, −3.66850945845351285121733608042, −3.35793653552807367647961636790, −3.33280814025082144925112783914, −3.10639230286477890234560324033, −2.79572318594033016940852819581, −2.07324358948568412247001315509, −1.81442019420303235171289493423, −1.75987996401026266995800847667, −0.58040493233256980285088634444, −0.19048608395910108070772185177,
0.19048608395910108070772185177, 0.58040493233256980285088634444, 1.75987996401026266995800847667, 1.81442019420303235171289493423, 2.07324358948568412247001315509, 2.79572318594033016940852819581, 3.10639230286477890234560324033, 3.33280814025082144925112783914, 3.35793653552807367647961636790, 3.66850945845351285121733608042, 4.24767152600730571291701615058, 4.79801644608287067850378744332, 4.90059560415257564000565627835, 5.04806526204044684207028691682, 5.51594457101220937413974190763, 5.59584836624783009560434669158, 5.90666910886784429580700605242, 5.99319272472723371802268190812, 6.60014490214433419660108498344, 6.69358011579204881157978676911, 6.74195158675404802760331752652, 7.13525026459992030418035652186, 7.51475251189716923085089632017, 7.87713074503043243135403273829, 8.043208572205063538089282139160
