| L(s) = 1 | − 4·3-s + 2·5-s − 4·7-s + 8·9-s − 2·13-s − 8·15-s − 10·17-s + 16·21-s − 4·23-s − 25-s − 12·27-s + 8·29-s − 8·35-s + 2·37-s + 8·39-s − 12·43-s + 16·45-s − 4·47-s + 8·49-s + 40·51-s + 14·53-s − 32·63-s − 4·65-s − 20·67-s + 16·69-s + 6·73-s + 4·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s − 1.51·7-s + 8/3·9-s − 0.554·13-s − 2.06·15-s − 2.42·17-s + 3.49·21-s − 0.834·23-s − 1/5·25-s − 2.30·27-s + 1.48·29-s − 1.35·35-s + 0.328·37-s + 1.28·39-s − 1.82·43-s + 2.38·45-s − 0.583·47-s + 8/7·49-s + 5.60·51-s + 1.92·53-s − 4.03·63-s − 0.496·65-s − 2.44·67-s + 1.92·69-s + 0.702·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.559149717192748233159057925916, −9.227162907626061810604438055305, −8.934634026978809288894843521812, −8.181617347687512346732753691470, −7.79982580989299237043573005582, −6.94560153214819664398373792850, −6.70473827940314624576708931637, −6.49277998876879542439994169626, −6.24731003724845975900784473318, −5.80160968803812973473268083478, −5.31943729848021380028926648088, −4.95900130163037522582916233958, −4.49959312532313205039525155716, −3.98858296705169004673274861768, −3.35946771613871041473133198147, −2.40800894228513447238391728865, −2.22746781042095738493630772532, −1.16129928007178687243288450535, 0, 0,
1.16129928007178687243288450535, 2.22746781042095738493630772532, 2.40800894228513447238391728865, 3.35946771613871041473133198147, 3.98858296705169004673274861768, 4.49959312532313205039525155716, 4.95900130163037522582916233958, 5.31943729848021380028926648088, 5.80160968803812973473268083478, 6.24731003724845975900784473318, 6.49277998876879542439994169626, 6.70473827940314624576708931637, 6.94560153214819664398373792850, 7.79982580989299237043573005582, 8.181617347687512346732753691470, 8.934634026978809288894843521812, 9.227162907626061810604438055305, 9.559149717192748233159057925916
