| L(s) = 1 | + 6·5-s − 6·11-s − 13-s − 6·17-s − 4·19-s + 6·23-s + 17·25-s + 3·29-s + 5·31-s − 2·37-s − 3·41-s + 43-s + 9·47-s − 6·53-s − 36·55-s + 3·59-s − 13·61-s − 6·65-s + 7·67-s + 24·71-s − 10·73-s − 11·79-s + 9·83-s − 36·85-s − 6·89-s − 24·95-s + 11·97-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 17/5·25-s + 0.557·29-s + 0.898·31-s − 0.328·37-s − 0.468·41-s + 0.152·43-s + 1.31·47-s − 0.824·53-s − 4.85·55-s + 0.390·59-s − 1.66·61-s − 0.744·65-s + 0.855·67-s + 2.84·71-s − 1.17·73-s − 1.23·79-s + 0.987·83-s − 3.90·85-s − 0.635·89-s − 2.46·95-s + 1.11·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.319291784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.319291784\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.727983819894462360005376025673, −7.989963590901599018360246155508, −7.59138198208800436072287462643, −7.32937667828391958558298336383, −6.75703296297873166917452565292, −6.40123330487083928930825491305, −6.26729085358700877029278918108, −5.94734211382088224877077564373, −5.43595382165398384515773190355, −5.08281092781559029470218025238, −4.89552545801240067668619757131, −4.61782232909975529374115898003, −3.95909531820594637158109832633, −3.29515807268657889379469506677, −2.74961780244005007125561255462, −2.47725415776011799234868648549, −2.22224054598588813593369546812, −1.86008327995803345363273322274, −1.22122199731996706812437831191, −0.45649960206435032937598750492,
0.45649960206435032937598750492, 1.22122199731996706812437831191, 1.86008327995803345363273322274, 2.22224054598588813593369546812, 2.47725415776011799234868648549, 2.74961780244005007125561255462, 3.29515807268657889379469506677, 3.95909531820594637158109832633, 4.61782232909975529374115898003, 4.89552545801240067668619757131, 5.08281092781559029470218025238, 5.43595382165398384515773190355, 5.94734211382088224877077564373, 6.26729085358700877029278918108, 6.40123330487083928930825491305, 6.75703296297873166917452565292, 7.32937667828391958558298336383, 7.59138198208800436072287462643, 7.989963590901599018360246155508, 8.727983819894462360005376025673
