L(s) = 1 | − 96·25-s − 196·49-s − 384·73-s − 576·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | − 3.83·25-s − 4·49-s − 5.26·73-s − 5.93·97-s − 4·121-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 + 2.81·169-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 + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07663744112\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07663744112\) |
\(L(2)\) |
|
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2}( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 5040 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 12480 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{4} \) |
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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
−6.73409262861747162276703595520, −6.65956751128351391715288950964, −6.44132915302646410752280583769, −5.88362042695444301390147581064, −5.86025589245632557193356903111, −5.80169330441415293404076923427, −5.69679986082384336136700266461, −5.22926650391183677909339365195, −4.89493549461387179182266966862, −4.76472020809121461654195407482, −4.61146326919192269816293046015, −4.08984967877816730073850536097, −3.92487389608208081990526233590, −3.87691394928541556837476485420, −3.73110585597308865210701653239, −3.03732579121269054906311726144, −2.93502635702133438127901364309, −2.80288814735066651683582761967, −2.47221788160497823330499447943, −1.91588778574575743945702042982, −1.58805408817701538525592718773, −1.54808679605004454174135947935, −1.36438076980014877138084177606, −0.41873058858541769324935210586, −0.05832625287861352165766350484,
0.05832625287861352165766350484, 0.41873058858541769324935210586, 1.36438076980014877138084177606, 1.54808679605004454174135947935, 1.58805408817701538525592718773, 1.91588778574575743945702042982, 2.47221788160497823330499447943, 2.80288814735066651683582761967, 2.93502635702133438127901364309, 3.03732579121269054906311726144, 3.73110585597308865210701653239, 3.87691394928541556837476485420, 3.92487389608208081990526233590, 4.08984967877816730073850536097, 4.61146326919192269816293046015, 4.76472020809121461654195407482, 4.89493549461387179182266966862, 5.22926650391183677909339365195, 5.69679986082384336136700266461, 5.80169330441415293404076923427, 5.86025589245632557193356903111, 5.88362042695444301390147581064, 6.44132915302646410752280583769, 6.65956751128351391715288950964, 6.73409262861747162276703595520