| L(s) = 1 | + 2·9-s + 8·13-s − 24·37-s + 20·49-s − 8·61-s + 24·73-s − 5·81-s − 40·97-s − 72·109-s + 16·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.21·13-s − 3.94·37-s + 20/7·49-s − 1.02·61-s + 2.80·73-s − 5/9·81-s − 4.06·97-s − 6.89·109-s + 1.47·117-s − 2.54·121-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 − 0.923·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303982652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303982652\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.52110195410064776192899065615, −6.01028574904119782214331278579, −5.92956718010253861627995489584, −5.84621947600543886534596801369, −5.46893557662991621026037465272, −5.29589590278807612455507150212, −5.21040680626613293114808581235, −4.99808145327271199441777000039, −4.71530273667890084231236815713, −4.44762367533319046171868414024, −4.00561398072340331187026887735, −3.86008397167915270229148299881, −3.73000800221805211686344178897, −3.71426032775093974215492601380, −3.69659114098635624243310081668, −2.91417210349562252921681796758, −2.79325014820246496794159047774, −2.61928168166826222247937395094, −2.46163869772699802124679945956, −1.80456602442220595474995627192, −1.58584850436301802763179010709, −1.38704411029660159039835411193, −1.36855541349147096405824818904, −0.75387623174760378231378370716, −0.25400413849612438516472408089,
0.25400413849612438516472408089, 0.75387623174760378231378370716, 1.36855541349147096405824818904, 1.38704411029660159039835411193, 1.58584850436301802763179010709, 1.80456602442220595474995627192, 2.46163869772699802124679945956, 2.61928168166826222247937395094, 2.79325014820246496794159047774, 2.91417210349562252921681796758, 3.69659114098635624243310081668, 3.71426032775093974215492601380, 3.73000800221805211686344178897, 3.86008397167915270229148299881, 4.00561398072340331187026887735, 4.44762367533319046171868414024, 4.71530273667890084231236815713, 4.99808145327271199441777000039, 5.21040680626613293114808581235, 5.29589590278807612455507150212, 5.46893557662991621026037465272, 5.84621947600543886534596801369, 5.92956718010253861627995489584, 6.01028574904119782214331278579, 6.52110195410064776192899065615
