| L(s) = 1 | + 4·13-s − 16·19-s + 28·31-s − 8·37-s − 28·43-s + 10·49-s + 40·61-s − 4·67-s − 8·79-s − 28·97-s + 40·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 3.67·19-s + 5.02·31-s − 1.31·37-s − 4.26·43-s + 10/7·49-s + 5.12·61-s − 0.488·67-s − 0.900·79-s − 2.84·97-s + 3.83·109-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 + 8/13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.630059844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.630059844\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 31 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 233 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 5407 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 6638 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 11753 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 176 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
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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
−6.57047038185955869264834169307, −6.51474285188188477649224482213, −6.44965913671592495934690055047, −6.09673628783353262566640322471, −5.76168320661378781797843009014, −5.61679998911892396901549475945, −5.32203705537297930576283329364, −5.20696105177361873452597887825, −4.72470743916973624722102897566, −4.61617825429957439785113011286, −4.52651028049774019182629755824, −4.15948375272246200102832971058, −4.13206295393141983410546048368, −3.70691621636216182264198191133, −3.58322101704458874188919770058, −3.15682121470461004670545082233, −3.08391771379898257069992145535, −2.58420989633150129128587210485, −2.43742896159130740677269104328, −2.21278590021175503535636268027, −1.74001889991787162819173362050, −1.70448056425791374423327413923, −1.17039107782987045162255944288, −0.57172767480076682605754449970, −0.52282341550348508225363999866,
0.52282341550348508225363999866, 0.57172767480076682605754449970, 1.17039107782987045162255944288, 1.70448056425791374423327413923, 1.74001889991787162819173362050, 2.21278590021175503535636268027, 2.43742896159130740677269104328, 2.58420989633150129128587210485, 3.08391771379898257069992145535, 3.15682121470461004670545082233, 3.58322101704458874188919770058, 3.70691621636216182264198191133, 4.13206295393141983410546048368, 4.15948375272246200102832971058, 4.52651028049774019182629755824, 4.61617825429957439785113011286, 4.72470743916973624722102897566, 5.20696105177361873452597887825, 5.32203705537297930576283329364, 5.61679998911892396901549475945, 5.76168320661378781797843009014, 6.09673628783353262566640322471, 6.44965913671592495934690055047, 6.51474285188188477649224482213, 6.57047038185955869264834169307
