| L(s) = 1 | + 2·9-s − 12·17-s + 12·41-s − 14·49-s − 4·73-s − 5·81-s + 36·89-s + 20·97-s − 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.91·17-s + 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s + 3.81·89-s + 2.03·97-s − 3.38·113-s − 1.27·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.268998693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268998693\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.004604059297186575675466799584, −7.969927599294231266677202823616, −7.47739101701819609266179145244, −7.06141323459836736777381614098, −6.86190371842249173263160904110, −6.37686032415289551578984947366, −6.11153633976116079454683525069, −6.02842313016901507347653475403, −5.06387741436896575651738877777, −4.96673371219222678923271082352, −4.69136050196791769810329408373, −4.15213607452696720572799923352, −3.92955226986834837112672681695, −3.54584521107838647336478443335, −2.77015956688253651923819653440, −2.59010901828974344812434161911, −2.03809384289219825909719242311, −1.70611701145266943750242581733, −1.04057642387582761234161837251, −0.30036950091384718759111670058,
0.30036950091384718759111670058, 1.04057642387582761234161837251, 1.70611701145266943750242581733, 2.03809384289219825909719242311, 2.59010901828974344812434161911, 2.77015956688253651923819653440, 3.54584521107838647336478443335, 3.92955226986834837112672681695, 4.15213607452696720572799923352, 4.69136050196791769810329408373, 4.96673371219222678923271082352, 5.06387741436896575651738877777, 6.02842313016901507347653475403, 6.11153633976116079454683525069, 6.37686032415289551578984947366, 6.86190371842249173263160904110, 7.06141323459836736777381614098, 7.47739101701819609266179145244, 7.969927599294231266677202823616, 8.004604059297186575675466799584
