| L(s) = 1 | − 6·5-s + 5·9-s + 19·25-s − 14·37-s − 30·45-s − 28·49-s + 24·53-s + 9·81-s + 18·89-s + 34·97-s + 42·113-s − 22·121-s − 66·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 84·185-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 5/3·9-s + 19/5·25-s − 2.30·37-s − 4.47·45-s − 4·49-s + 3.29·53-s + 81-s + 1.90·89-s + 3.45·97-s + 3.95·113-s − 2·121-s − 5.90·125-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 6.17·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7555589859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7555589859\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} )( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.212996674046203735337311202936, −8.971505389799373493866520546261, −8.655670082023741286117798766968, −8.545077866047027119965392549875, −8.071198135635803455080224758825, −8.048220585049429317601319487011, −7.55691026273009723830964780581, −7.36446723234155150585542721975, −7.22633915248444376692318463132, −7.14094660299034634491094924629, −6.47809816408198717520290975700, −6.31941455450258019569995334222, −6.26603663170649072035034793944, −5.18262010759134743285302351250, −5.12318679299002574848960217972, −5.09749591446562663117408713672, −4.40329466957152938385690243502, −4.19460063633393502825970530886, −4.00216064520592543125067253567, −3.46856806928306764618609269897, −3.42229060141302357500580330943, −2.96037288246730678304538456970, −2.07051252179643636702352886208, −1.62860371650146495490960633779, −0.63556781156953398316093940531,
0.63556781156953398316093940531, 1.62860371650146495490960633779, 2.07051252179643636702352886208, 2.96037288246730678304538456970, 3.42229060141302357500580330943, 3.46856806928306764618609269897, 4.00216064520592543125067253567, 4.19460063633393502825970530886, 4.40329466957152938385690243502, 5.09749591446562663117408713672, 5.12318679299002574848960217972, 5.18262010759134743285302351250, 6.26603663170649072035034793944, 6.31941455450258019569995334222, 6.47809816408198717520290975700, 7.14094660299034634491094924629, 7.22633915248444376692318463132, 7.36446723234155150585542721975, 7.55691026273009723830964780581, 8.048220585049429317601319487011, 8.071198135635803455080224758825, 8.545077866047027119965392549875, 8.655670082023741286117798766968, 8.971505389799373493866520546261, 9.212996674046203735337311202936
