| L(s) = 1 | − 16·13-s + 16·25-s + 24·37-s − 4·49-s + 8·61-s − 32·97-s − 48·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 16/5·25-s + 3.94·37-s − 4/7·49-s + 1.02·61-s − 3.24·97-s − 4.59·109-s + 1.81·121-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 + 8.30·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.314110566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314110566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−7.64447280433074367089497536340, −7.64006565909740857192430892332, −7.15114571207234710370819670074, −7.07012598023383490879371606100, −6.96603639246896978904741026584, −6.52109639569291087815479229536, −6.41109217734208565932017327079, −6.18668713965497884219095361266, −5.59283793293401047972675320564, −5.47682241251627425783245967948, −5.27264475672384624893226264143, −4.81626122090875509237041380955, −4.80126549567677412640917389036, −4.70885502241919532450136180327, −4.30200732071547033856468519348, −3.98491503100421461092184165467, −3.76218282389824909556386566466, −2.87620669905022926389970862274, −2.83355167419338710386771526045, −2.82630491320167199391182977328, −2.47865409445110051051012942284, −2.20165118443037188677233009361, −1.51386925148197022456047979947, −1.00139004576048210737172999504, −0.39560713202295222952378087818,
0.39560713202295222952378087818, 1.00139004576048210737172999504, 1.51386925148197022456047979947, 2.20165118443037188677233009361, 2.47865409445110051051012942284, 2.82630491320167199391182977328, 2.83355167419338710386771526045, 2.87620669905022926389970862274, 3.76218282389824909556386566466, 3.98491503100421461092184165467, 4.30200732071547033856468519348, 4.70885502241919532450136180327, 4.80126549567677412640917389036, 4.81626122090875509237041380955, 5.27264475672384624893226264143, 5.47682241251627425783245967948, 5.59283793293401047972675320564, 6.18668713965497884219095361266, 6.41109217734208565932017327079, 6.52109639569291087815479229536, 6.96603639246896978904741026584, 7.07012598023383490879371606100, 7.15114571207234710370819670074, 7.64006565909740857192430892332, 7.64447280433074367089497536340
