| L(s) = 1 | − 4·4-s + 14·9-s + 28·11-s + 16·16-s + 68·19-s − 56·36-s − 92·41-s − 112·44-s − 98·49-s + 164·59-s − 64·64-s − 272·76-s + 115·81-s − 292·89-s + 392·99-s + 346·121-s + 127-s + 131-s + 137-s + 139-s + 224·144-s + 149-s + 151-s + 157-s + 163-s + 368·164-s + 167-s + ⋯ |
| L(s) = 1 | − 4-s + 14/9·9-s + 2.54·11-s + 16-s + 3.57·19-s − 1.55·36-s − 2.24·41-s − 2.54·44-s − 2·49-s + 2.77·59-s − 64-s − 3.57·76-s + 1.41·81-s − 3.28·89-s + 3.95·99-s + 2.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 14/9·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 2.24·164-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.347484676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347484676\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 574 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3502 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 5134 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9506 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 11186 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9982 T^{2} + p^{4} T^{4} \) |
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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
−12.45535388448681052119875677560, −11.91912971642635669010375862873, −11.62772873988172058288943148203, −11.28240271681750044573830856718, −10.04960361548778878615843955639, −9.960684830439755343682162171992, −9.573421795157107219679870960615, −9.207416856001995342023207358569, −8.563034476365096442179532645934, −7.992619588803758628425075456067, −7.10104945420974827463258993897, −7.09040272499742409114993402992, −6.35214899720827533830964109171, −5.42056447110308954452739864582, −5.02618562372392492258651330909, −4.26738074440473092841067019995, −3.67188807235135694849756694645, −3.31841546416268888117465483616, −1.38007531074061587277584289792, −1.19655335092803561406754300994,
1.19655335092803561406754300994, 1.38007531074061587277584289792, 3.31841546416268888117465483616, 3.67188807235135694849756694645, 4.26738074440473092841067019995, 5.02618562372392492258651330909, 5.42056447110308954452739864582, 6.35214899720827533830964109171, 7.09040272499742409114993402992, 7.10104945420974827463258993897, 7.992619588803758628425075456067, 8.563034476365096442179532645934, 9.207416856001995342023207358569, 9.573421795157107219679870960615, 9.960684830439755343682162171992, 10.04960361548778878615843955639, 11.28240271681750044573830856718, 11.62772873988172058288943148203, 11.91912971642635669010375862873, 12.45535388448681052119875677560
