| L(s) = 1 | + 8·9-s + 12·25-s − 24·41-s − 12·49-s + 48·73-s + 30·81-s + 48·89-s − 32·97-s + 24·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 12/5·25-s − 3.74·41-s − 1.71·49-s + 5.61·73-s + 10/3·81-s + 5.08·89-s − 3.24·97-s + 2.25·113-s + 8/11·121-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 − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.701787816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.701787816\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−7.79909552216607678444884477292, −7.71694690159245504258244220662, −7.32274732681820009610423393563, −7.04690214289566320262486538611, −6.88486214318876501504601253388, −6.66093633969141580322601428520, −6.54713293326644536229804305125, −6.36456944693504323952206829025, −6.13958336556772570864843082286, −5.30201588502053273568400954162, −5.23615319437143790446932754217, −5.20188634365607806817631801776, −4.91785648967475568769244281757, −4.45613590733488512215061262159, −4.45157037610707154806834886897, −4.09059378734539488942334858323, −3.55976205423273291896366405874, −3.37764736515888330053560081784, −3.33799023315942635708491927443, −2.83498155362885020650890898722, −2.13825128632969348201908881274, −1.92277358369078056138199127899, −1.73372183621146676023011668910, −1.07175944957755366080494473610, −0.74451206334897596803652742335,
0.74451206334897596803652742335, 1.07175944957755366080494473610, 1.73372183621146676023011668910, 1.92277358369078056138199127899, 2.13825128632969348201908881274, 2.83498155362885020650890898722, 3.33799023315942635708491927443, 3.37764736515888330053560081784, 3.55976205423273291896366405874, 4.09059378734539488942334858323, 4.45157037610707154806834886897, 4.45613590733488512215061262159, 4.91785648967475568769244281757, 5.20188634365607806817631801776, 5.23615319437143790446932754217, 5.30201588502053273568400954162, 6.13958336556772570864843082286, 6.36456944693504323952206829025, 6.54713293326644536229804305125, 6.66093633969141580322601428520, 6.88486214318876501504601253388, 7.04690214289566320262486538611, 7.32274732681820009610423393563, 7.71694690159245504258244220662, 7.79909552216607678444884477292
