| L(s) = 1 | − 8·7-s − 9-s − 12·17-s + 6·25-s − 8·31-s − 4·41-s − 16·47-s + 34·49-s + 8·63-s − 32·71-s + 12·73-s − 8·79-s + 81-s − 20·89-s − 28·97-s − 24·103-s + 4·113-s + 96·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/3·9-s − 2.91·17-s + 6/5·25-s − 1.43·31-s − 0.624·41-s − 2.33·47-s + 34/7·49-s + 1.00·63-s − 3.79·71-s + 1.40·73-s − 0.900·79-s + 1/9·81-s − 2.11·89-s − 2.84·97-s − 2.36·103-s + 0.376·113-s + 8.80·119-s + 6/11·121-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 + 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.992657737159952110923654674533, −9.687819352404895063426047923888, −9.199150600618356363333291423906, −8.998327639266284776293447036244, −8.605948567033838895002997751855, −8.109292621001179927364041734470, −7.06992965408008235742663108340, −7.02636487220597968841948174938, −6.55420735276939294351881900748, −6.43427622228386769543730958206, −5.73167023113259948249081162165, −5.34799277350386073726216311847, −4.32137562834720234205738904062, −4.29370400578789576461250993928, −3.33072250231337258947255629984, −3.06310481619601238631488333900, −2.61280563257824176573263690653, −1.73207908456882757664741684628, 0, 0,
1.73207908456882757664741684628, 2.61280563257824176573263690653, 3.06310481619601238631488333900, 3.33072250231337258947255629984, 4.29370400578789576461250993928, 4.32137562834720234205738904062, 5.34799277350386073726216311847, 5.73167023113259948249081162165, 6.43427622228386769543730958206, 6.55420735276939294351881900748, 7.02636487220597968841948174938, 7.06992965408008235742663108340, 8.109292621001179927364041734470, 8.605948567033838895002997751855, 8.998327639266284776293447036244, 9.199150600618356363333291423906, 9.687819352404895063426047923888, 9.992657737159952110923654674533
