| L(s) = 1 | + 576·19-s − 400·25-s + 220·49-s − 2.88e3·67-s + 2.24e3·73-s + 800·97-s + 3.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 6.95·19-s − 3.19·25-s + 0.641·49-s − 5.25·67-s + 3.59·73-s + 0.837·97-s + 2.26·121-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 + 3.70·169-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 + 0.000300·223-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.54691766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.54691766\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 8 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1510 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2}( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 5776 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 13966 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 34328 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 45182 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 100010 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 124720 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 51554 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 296696 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 244870 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 356038 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 456622 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 560 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 856478 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 423574 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 449440 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{4} \) |
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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
−6.75937601542700854865859841933, −6.19716020389062558865497168909, −5.96110016086126613657225525919, −5.94847303298493532538879598247, −5.62439185236494804325640454031, −5.53240829610530151043703832577, −5.48956075929587626217948713574, −4.98750278317160644358054291036, −4.80353992606234855298208147241, −4.72893667498353845913786914524, −4.30858332742626482112700808606, −4.02483260132629946638667457630, −3.67839502870794857571299537061, −3.39720317336625848190699452471, −3.24410183034831473367828683324, −3.23237856615943452605737053022, −2.95854619416337246643642642414, −2.38211741446302018823616515011, −2.30432169697726204873526010106, −1.67816299804933445944084167308, −1.48696434013285558741463937094, −1.38912687925980251698121633225, −0.834023677192400467581231616096, −0.53078282530583956709469459158, −0.48911610427246907416785565613,
0.48911610427246907416785565613, 0.53078282530583956709469459158, 0.834023677192400467581231616096, 1.38912687925980251698121633225, 1.48696434013285558741463937094, 1.67816299804933445944084167308, 2.30432169697726204873526010106, 2.38211741446302018823616515011, 2.95854619416337246643642642414, 3.23237856615943452605737053022, 3.24410183034831473367828683324, 3.39720317336625848190699452471, 3.67839502870794857571299537061, 4.02483260132629946638667457630, 4.30858332742626482112700808606, 4.72893667498353845913786914524, 4.80353992606234855298208147241, 4.98750278317160644358054291036, 5.48956075929587626217948713574, 5.53240829610530151043703832577, 5.62439185236494804325640454031, 5.94847303298493532538879598247, 5.96110016086126613657225525919, 6.19716020389062558865497168909, 6.75937601542700854865859841933
