| L(s) = 1 | − 8·16-s − 18·23-s + 25-s − 14·37-s + 24·47-s + 12·53-s + 26·67-s + 12·71-s − 34·97-s + 8·103-s + 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2·16-s − 3.75·23-s + 1/5·25-s − 2.30·37-s + 3.50·47-s + 1.64·53-s + 3.17·67-s + 1.42·71-s − 3.45·97-s + 0.788·103-s + 3.95·113-s + 2·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074572780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074572780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} ) \) |
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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
−7.889812421802828281950861730757, −7.64815083755940574382124367817, −7.42377249847246443719676892318, −7.15272354519761523863634944298, −6.88760561470218143496508899027, −6.69540845508282376977708910248, −6.65376058647720259342819785698, −6.14999519712168288598715564797, −5.81890170407889759323750397280, −5.68587382440119764255050413021, −5.61177083109721757128607903966, −5.10550727424362354083474764831, −4.96966564952909587647106592625, −4.42667115387506240720446252366, −4.20029597274579801027974039225, −4.17935379818873751739239735387, −3.80130001666780003834059531876, −3.47405031030584550676182254106, −3.24671931664923696691218546011, −2.49581096514771664488724975540, −2.27457280885841277135217566197, −2.07804869394233231739161269503, −1.97773268211154975998856887080, −1.05182651121211243436205258059, −0.37048664922876438918029661543,
0.37048664922876438918029661543, 1.05182651121211243436205258059, 1.97773268211154975998856887080, 2.07804869394233231739161269503, 2.27457280885841277135217566197, 2.49581096514771664488724975540, 3.24671931664923696691218546011, 3.47405031030584550676182254106, 3.80130001666780003834059531876, 4.17935379818873751739239735387, 4.20029597274579801027974039225, 4.42667115387506240720446252366, 4.96966564952909587647106592625, 5.10550727424362354083474764831, 5.61177083109721757128607903966, 5.68587382440119764255050413021, 5.81890170407889759323750397280, 6.14999519712168288598715564797, 6.65376058647720259342819785698, 6.69540845508282376977708910248, 6.88760561470218143496508899027, 7.15272354519761523863634944298, 7.42377249847246443719676892318, 7.64815083755940574382124367817, 7.889812421802828281950861730757
