| L(s) = 1 | − 20·7-s − 28·13-s + 4·19-s − 10·25-s + 28·31-s − 64·37-s − 116·43-s + 144·49-s − 4·61-s − 56·67-s − 28·73-s + 268·79-s + 560·91-s + 8·97-s + 400·103-s + 140·109-s + 376·121-s + 127-s + 131-s − 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.85·7-s − 2.15·13-s + 4/19·19-s − 2/5·25-s + 0.903·31-s − 1.72·37-s − 2.69·43-s + 2.93·49-s − 0.0655·61-s − 0.835·67-s − 0.383·73-s + 3.39·79-s + 6.15·91-s + 8/97·97-s + 3.88·103-s + 1.28·109-s + 3.10·121-s + 0.00787·127-s + 0.00763·131-s − 0.601·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2648810944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2648810944\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 10 T + 78 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 376 T^{2} + 63006 T^{4} - 376 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 14 T + 342 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 353 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 543 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 38 p T^{2} + 433131 T^{4} - 38 p^{5} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3256 T^{2} + 4063326 T^{4} - 3256 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 14 T + 1791 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 32 T + 1374 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4312 T^{2} + 9935358 T^{4} - 4312 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 58 T + 2334 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8188 T^{2} + 26416518 T^{4} - 8188 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2218 T^{2} + 16848843 T^{4} - 2218 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7912 T^{2} + 35671038 T^{4} - 7912 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 963 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11704 T^{2} + 67208766 T^{4} - 11704 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 14 T + 10302 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 134 T + 10491 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2338 T^{2} + 46102083 T^{4} - 2338 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 5512 T^{2} + 128866398 T^{4} - 5512 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 18642 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.15520337940245998411410790506, −6.11975227575004077631893119551, −5.95303973762374004943388819176, −5.93844323699500108986644232054, −5.30132916203829281385766317978, −5.06225181883497336214245021633, −4.96173449225333657528992891265, −4.80772815357307259725800855289, −4.74801411947112610582821577092, −4.47551953854594537920016044272, −3.92028767413694705581123786683, −3.64370874083565214362974430099, −3.48761474003375215977065483819, −3.45209848100669186490158424148, −3.36019647639628026039147381361, −2.95447763617178436020835607191, −2.55541853993553361180115240354, −2.51438944098647962286301159526, −2.25443585167870867333135640754, −1.96931647892790466040500233570, −1.49956988696577790762575541113, −1.33078258958841618537562736342, −0.57920948154722457338545198220, −0.48805469851330721458364115266, −0.11225567927891783844417251355,
0.11225567927891783844417251355, 0.48805469851330721458364115266, 0.57920948154722457338545198220, 1.33078258958841618537562736342, 1.49956988696577790762575541113, 1.96931647892790466040500233570, 2.25443585167870867333135640754, 2.51438944098647962286301159526, 2.55541853993553361180115240354, 2.95447763617178436020835607191, 3.36019647639628026039147381361, 3.45209848100669186490158424148, 3.48761474003375215977065483819, 3.64370874083565214362974430099, 3.92028767413694705581123786683, 4.47551953854594537920016044272, 4.74801411947112610582821577092, 4.80772815357307259725800855289, 4.96173449225333657528992891265, 5.06225181883497336214245021633, 5.30132916203829281385766317978, 5.93844323699500108986644232054, 5.95303973762374004943388819176, 6.11975227575004077631893119551, 6.15520337940245998411410790506
