| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s − 4·16-s + 12·17-s − 96·23-s + 52·31-s + 32·32-s − 48·34-s + 384·46-s − 108·47-s − 84·53-s − 100·61-s − 208·62-s − 64·64-s + 96·68-s − 276·83-s − 768·92-s + 432·94-s + 336·106-s − 504·107-s − 360·113-s − 376·121-s + 400·122-s + 416·124-s + 127-s + 64·128-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s − 8-s − 1/4·16-s + 0.705·17-s − 4.17·23-s + 1.67·31-s + 32-s − 1.41·34-s + 8.34·46-s − 2.29·47-s − 1.58·53-s − 1.63·61-s − 3.35·62-s − 64-s + 1.41·68-s − 3.32·83-s − 8.34·92-s + 4.59·94-s + 3.16·106-s − 4.71·107-s − 3.18·113-s − 3.10·121-s + 3.27·122-s + 3.35·124-s + 0.00787·127-s + 1/2·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03373786207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03373786207\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 + 4223 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 188 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1628 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 1892303 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 3308 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 3344879 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 42 T + 882 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 814 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 23793794 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 4682 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 51819121 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 10273 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 138 T + 9522 T^{2} + 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 5758 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 175995601 T^{4} + p^{8} T^{8} \) |
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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
−6.67083312020295229508532694660, −6.57801043410515697506408476044, −6.45772035696245391556853830478, −6.05477710492443966965718868983, −5.83808232866367780239511432921, −5.70845478114787486542601941400, −5.52442445681041412952464684208, −5.03739474171118247389242853114, −4.78780051996292350191265002404, −4.78212826197223224747279151112, −4.39215871756365757725768297915, −4.03942626076878687095187778959, −3.85892229299931095741797758595, −3.63591704780140960285453561590, −3.58393131905716022971985993966, −2.79972644917705374615658724174, −2.72832739062195903183919595991, −2.46757535357931043918470603473, −2.42923992404305411061497257855, −1.60241978854852296272809778438, −1.52338526406691616798159036179, −1.34096497053802826007310967001, −1.25681599266906359024682898146, −0.23264056211249312938356497434, −0.097164270453935729573331896835,
0.097164270453935729573331896835, 0.23264056211249312938356497434, 1.25681599266906359024682898146, 1.34096497053802826007310967001, 1.52338526406691616798159036179, 1.60241978854852296272809778438, 2.42923992404305411061497257855, 2.46757535357931043918470603473, 2.72832739062195903183919595991, 2.79972644917705374615658724174, 3.58393131905716022971985993966, 3.63591704780140960285453561590, 3.85892229299931095741797758595, 4.03942626076878687095187778959, 4.39215871756365757725768297915, 4.78212826197223224747279151112, 4.78780051996292350191265002404, 5.03739474171118247389242853114, 5.52442445681041412952464684208, 5.70845478114787486542601941400, 5.83808232866367780239511432921, 6.05477710492443966965718868983, 6.45772035696245391556853830478, 6.57801043410515697506408476044, 6.67083312020295229508532694660
