| L(s) = 1 | − 8·23-s − 2·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + 257-s + ⋯ |
| L(s) = 1 | − 8·23-s − 2·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005688524006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005688524006\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + T )^{8} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.27419284648995961516511332131, −6.15796383668147331066746165520, −6.02324429688313352644139885608, −5.98259576386807989393472877714, −5.62337833200490643927355931486, −5.51306654170881316723314959061, −5.33042603206199091461793389858, −4.98161625492424237018264718628, −4.62217260578478365151197680884, −4.55276693275261043994045387141, −4.15819276354265520256157759950, −4.13748623149288309061355353688, −3.83748970608071710770929645148, −3.78910973679140441092664974012, −3.58915910873875885073778767583, −3.48302567746473968591422796013, −2.86182270781797951769350910301, −2.52694951038808277182534951823, −2.48241400734359867856245209807, −2.05243266777295900016439380181, −1.99485879519982795400567975238, −1.68080760019789721212063916576, −1.65650630355037550794186077972, −0.968097893712279132460483011850, −0.03051562360724626767281873836,
0.03051562360724626767281873836, 0.968097893712279132460483011850, 1.65650630355037550794186077972, 1.68080760019789721212063916576, 1.99485879519982795400567975238, 2.05243266777295900016439380181, 2.48241400734359867856245209807, 2.52694951038808277182534951823, 2.86182270781797951769350910301, 3.48302567746473968591422796013, 3.58915910873875885073778767583, 3.78910973679140441092664974012, 3.83748970608071710770929645148, 4.13748623149288309061355353688, 4.15819276354265520256157759950, 4.55276693275261043994045387141, 4.62217260578478365151197680884, 4.98161625492424237018264718628, 5.33042603206199091461793389858, 5.51306654170881316723314959061, 5.62337833200490643927355931486, 5.98259576386807989393472877714, 6.02324429688313352644139885608, 6.15796383668147331066746165520, 6.27419284648995961516511332131
