| L(s) = 1 | − 2·3-s − 4-s − 2·5-s − 3·9-s + 2·12-s + 4·15-s − 3·16-s + 2·20-s + 3·25-s + 14·27-s − 16·31-s + 3·36-s − 16·37-s + 6·45-s + 18·47-s + 6·48-s − 11·49-s + 12·53-s − 24·59-s − 4·60-s + 7·64-s − 10·67-s − 24·71-s − 6·75-s + 6·80-s − 4·81-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s − 9-s + 0.577·12-s + 1.03·15-s − 3/4·16-s + 0.447·20-s + 3/5·25-s + 2.69·27-s − 2.87·31-s + 1/2·36-s − 2.63·37-s + 0.894·45-s + 2.62·47-s + 0.866·48-s − 1.57·49-s + 1.64·53-s − 3.12·59-s − 0.516·60-s + 7/8·64-s − 1.22·67-s − 2.84·71-s − 0.692·75-s + 0.670·80-s − 4/9·81-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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
−10.81667643402762384082211745565, −10.24852761452030434688937769245, −9.329926879371495179584250281570, −9.097081782258434538096454156701, −8.696146579504793606544450782183, −8.449067249733353054667681702856, −7.70065205133643917795542983375, −7.12738424913006641520490750420, −7.05225309963815263333041362939, −6.21582976908259805341697702925, −5.69161695820892943497683929412, −5.51745327455328883867742473683, −4.86760448399064938279060773931, −4.48129188551813999289632652079, −3.75162009716497306769710738140, −3.28415683071600115352785166225, −2.57845760710912345016843008396, −1.53191520661976603474827760272, 0, 0,
1.53191520661976603474827760272, 2.57845760710912345016843008396, 3.28415683071600115352785166225, 3.75162009716497306769710738140, 4.48129188551813999289632652079, 4.86760448399064938279060773931, 5.51745327455328883867742473683, 5.69161695820892943497683929412, 6.21582976908259805341697702925, 7.05225309963815263333041362939, 7.12738424913006641520490750420, 7.70065205133643917795542983375, 8.449067249733353054667681702856, 8.696146579504793606544450782183, 9.097081782258434538096454156701, 9.329926879371495179584250281570, 10.24852761452030434688937769245, 10.81667643402762384082211745565
