| L(s) = 1 | − 4·4-s + 12·16-s − 20·25-s + 28·49-s − 32·64-s − 16·97-s + 80·100-s + 32·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 112·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 4·25-s + 4·49-s − 4·64-s − 1.62·97-s + 8·100-s + 3.06·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 8·196-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068433727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068433727\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−7.05407274032692323450092692887, −6.91000114602047133474098960636, −6.49988457234966172836817629412, −6.28212237291547466587029169893, −6.04570935010398604051007797096, −5.75608094301818517470278045781, −5.67494631280870866605629412617, −5.46429230084732328308210094454, −5.35801424630095609207171843217, −4.94666192547902837893206555797, −4.79358789200637832067230103975, −4.32147862298709959483989177862, −4.23136421316346341558510300069, −3.98989122780480665433544641159, −3.92307044648178206620584028391, −3.61444013622569900978559618053, −3.44153766615470922338977388915, −2.96582148288602374332611460310, −2.58546694277605090493116186104, −2.41349817698543257459087489629, −1.95287382278036221814286315035, −1.60617565673840333780245728411, −1.28090889499958778622894263624, −0.60374756455491515360145278190, −0.38058848589459349732142442157,
0.38058848589459349732142442157, 0.60374756455491515360145278190, 1.28090889499958778622894263624, 1.60617565673840333780245728411, 1.95287382278036221814286315035, 2.41349817698543257459087489629, 2.58546694277605090493116186104, 2.96582148288602374332611460310, 3.44153766615470922338977388915, 3.61444013622569900978559618053, 3.92307044648178206620584028391, 3.98989122780480665433544641159, 4.23136421316346341558510300069, 4.32147862298709959483989177862, 4.79358789200637832067230103975, 4.94666192547902837893206555797, 5.35801424630095609207171843217, 5.46429230084732328308210094454, 5.67494631280870866605629412617, 5.75608094301818517470278045781, 6.04570935010398604051007797096, 6.28212237291547466587029169893, 6.49988457234966172836817629412, 6.91000114602047133474098960636, 7.05407274032692323450092692887
