| L(s) = 1 | + 24·17-s + 116·25-s + 216·41-s + 164·49-s + 1.73e3·73-s + 4.72e3·89-s − 4.95e3·97-s − 6.60e3·113-s + 1.79e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.33e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.342·17-s + 0.927·25-s + 0.822·41-s + 0.478·49-s + 2.78·73-s + 5.63·89-s − 5.18·97-s − 5.49·113-s + 1.34·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.42·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3266406323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3266406323\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 898 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2666 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 4882 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 5134 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6710 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 47294 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 101114 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 36350 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 204574 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 292954 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 391714 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 269450 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 569842 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 22226 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 958430 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1070674 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1182 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1238 T + p^{3} T^{2} )^{4} \) |
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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
−7.27354999063674893914337077837, −6.93957732211319835574604653935, −6.78683740099675498220464056286, −6.71757344205006041229626385007, −6.42879048033109201072582584496, −6.08284798788883829550825678852, −5.76686767928619364065068675993, −5.68413717291761578564400274079, −5.33003076118195107657564546469, −5.00100516554523221824928620581, −4.87965823942515053499646583266, −4.72814698701715316749722775213, −4.09030356186910631260655469332, −3.96072169000761319483131538215, −3.92037044870045729715993562474, −3.33758843861140782091451793330, −3.19902945022969275854589225274, −2.82228954593286217053273787922, −2.44115526656448465811245005684, −2.21751610410192383334083112007, −1.95039949411512395902018752408, −1.23631755436381504838806195551, −1.09681044081014832834381554848, −0.822068487430762140239487582692, −0.07832095165807093530768169294,
0.07832095165807093530768169294, 0.822068487430762140239487582692, 1.09681044081014832834381554848, 1.23631755436381504838806195551, 1.95039949411512395902018752408, 2.21751610410192383334083112007, 2.44115526656448465811245005684, 2.82228954593286217053273787922, 3.19902945022969275854589225274, 3.33758843861140782091451793330, 3.92037044870045729715993562474, 3.96072169000761319483131538215, 4.09030356186910631260655469332, 4.72814698701715316749722775213, 4.87965823942515053499646583266, 5.00100516554523221824928620581, 5.33003076118195107657564546469, 5.68413717291761578564400274079, 5.76686767928619364065068675993, 6.08284798788883829550825678852, 6.42879048033109201072582584496, 6.71757344205006041229626385007, 6.78683740099675498220464056286, 6.93957732211319835574604653935, 7.27354999063674893914337077837
