| L(s) = 1 | − 48·13-s − 56·25-s − 16·37-s + 152·49-s + 112·61-s − 304·73-s + 112·97-s + 80·109-s + 520·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 840·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.69·13-s − 2.23·25-s − 0.432·37-s + 3.10·49-s + 1.83·61-s − 4.16·73-s + 1.15·97-s + 0.733·109-s + 4.29·121-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 + 4.97·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4555831849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4555831849\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 28 T^{2} + 22 p^{2} T^{4} + 28 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 76 T^{2} + 5350 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 130 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 12 T + 150 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 868 T^{2} + 341062 T^{4} + 868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 708 T^{2} + 256934 T^{4} - 708 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 196 T^{2} - 348218 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2524 T^{2} + 2963302 T^{4} + 2524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3084 T^{2} + 4152230 T^{4} - 3084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 2518 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3364 T^{2} + 7778182 T^{4} + 3364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 6148 T^{2} + 15928678 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 252 T^{2} + 1517702 T^{4} + 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4060 T^{2} + 11815462 T^{4} + 4060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8932 T^{2} + 40509862 T^{4} - 8932 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 28 T + 7414 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 676 T^{2} + 21837030 T^{4} - 676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 76 T + 4038 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 17228 T^{2} + 145319334 T^{4} - 17228 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 10948 T^{2} + 121211302 T^{4} - 10948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 11076 T^{2} + 54999110 T^{4} + 11076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 28 T + 15430 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.95664709705934268226455568436, −3.89984333124627846641639954128, −3.80355348241842356032215426317, −3.59198136158319168432289017025, −3.33691930642946646864597685247, −3.26771843814622011404055368035, −3.16474214210754996404059777030, −3.03587143546808325986171727246, −2.78570154666339365674711014882, −2.78208811227511670578316613928, −2.51818924908901280396714366303, −2.41469028844723940168996014708, −2.40549194627889394573290643809, −2.10416955409844456043443902020, −2.01690115704944984086888415461, −1.96197245887590614566181869230, −1.78854193779582947590308629225, −1.55094491000372837063118352376, −1.41673740643647942939761569129, −1.16275976667891159222666320054, −0.806345565025790125134213096417, −0.60465392801843076550749663476, −0.58499728805247873210903895453, −0.35384518659886570363726421658, −0.06465180664600223555494092243,
0.06465180664600223555494092243, 0.35384518659886570363726421658, 0.58499728805247873210903895453, 0.60465392801843076550749663476, 0.806345565025790125134213096417, 1.16275976667891159222666320054, 1.41673740643647942939761569129, 1.55094491000372837063118352376, 1.78854193779582947590308629225, 1.96197245887590614566181869230, 2.01690115704944984086888415461, 2.10416955409844456043443902020, 2.40549194627889394573290643809, 2.41469028844723940168996014708, 2.51818924908901280396714366303, 2.78208811227511670578316613928, 2.78570154666339365674711014882, 3.03587143546808325986171727246, 3.16474214210754996404059777030, 3.26771843814622011404055368035, 3.33691930642946646864597685247, 3.59198136158319168432289017025, 3.80355348241842356032215426317, 3.89984333124627846641639954128, 3.95664709705934268226455568436
Plot not available for L-functions of degree greater than 10.
