| L(s) = 1 | + 2.25e34·3-s + 2.23e43·4-s − 8.78e60·7-s + 5.07e68·9-s + 5.02e77·12-s − 2.98e80·13-s + 4.97e86·16-s − 1.55e92·19-s − 1.97e95·21-s + 4.48e100·25-s + 1.14e103·27-s − 1.95e104·28-s − 4.19e107·31-s + 1.13e112·36-s + 1.22e113·37-s − 6.71e114·39-s + 7.93e117·43-s + 1.12e121·48-s + 2.77e121·49-s − 6.64e123·52-s − 3.51e126·57-s − 6.85e128·61-s − 4.45e129·63-s + 1.10e130·64-s + 4.54e131·67-s − 1.34e134·73-s + 1.01e135·75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 1.24·7-s + 9-s + 12-s − 1.86·13-s + 16-s − 1.32·19-s − 1.24·21-s + 25-s + 27-s − 1.24·28-s − 1.75·31-s + 36-s + 1.50·37-s − 1.86·39-s + 1.95·43-s + 48-s + 0.560·49-s − 1.86·52-s − 1.32·57-s − 1.96·61-s − 1.24·63-s + 64-s + 1.51·67-s − 0.930·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(145-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+72) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{145}{2})\) |
\(\approx\) |
\(3.341125178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.341125178\) |
| \(L(73)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{72} T \) |
| good | 2 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 5 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 7 | \( 1 + \)\(87\!\cdots\!98\)\( T + p^{144} T^{2} \) |
| 11 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 13 | \( 1 + \)\(29\!\cdots\!38\)\( T + p^{144} T^{2} \) |
| 17 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 19 | \( 1 + \)\(15\!\cdots\!78\)\( T + p^{144} T^{2} \) |
| 23 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 29 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 31 | \( 1 + \)\(41\!\cdots\!78\)\( T + p^{144} T^{2} \) |
| 37 | \( 1 - \)\(12\!\cdots\!62\)\( T + p^{144} T^{2} \) |
| 41 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 43 | \( 1 - \)\(79\!\cdots\!02\)\( T + p^{144} T^{2} \) |
| 47 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 53 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 59 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 61 | \( 1 + \)\(68\!\cdots\!58\)\( T + p^{144} T^{2} \) |
| 67 | \( 1 - \)\(45\!\cdots\!22\)\( T + p^{144} T^{2} \) |
| 71 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 73 | \( 1 + \)\(13\!\cdots\!58\)\( T + p^{144} T^{2} \) |
| 79 | \( 1 - \)\(25\!\cdots\!82\)\( T + p^{144} T^{2} \) |
| 83 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 89 | \( ( 1 - p^{72} T )( 1 + p^{72} T ) \) |
| 97 | \( 1 - \)\(16\!\cdots\!82\)\( T + p^{144} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.876452868531820417698611293589, −9.061799797016799027523226967452, −7.63062562852736955739523278890, −7.03099495383801267957191478722, −6.05814485467362397135562784278, −4.53070506207337133386139595279, −3.35707964757651772467240795493, −2.58052323630328628991180298808, −2.01807168745358311355770646619, −0.59035658748351937724105266903,
0.59035658748351937724105266903, 2.01807168745358311355770646619, 2.58052323630328628991180298808, 3.35707964757651772467240795493, 4.53070506207337133386139595279, 6.05814485467362397135562784278, 7.03099495383801267957191478722, 7.63062562852736955739523278890, 9.061799797016799027523226967452, 9.876452868531820417698611293589
