| L(s) = 1 | − 2·3-s − 4-s + 4·5-s + 3·9-s + 4·11-s + 2·12-s − 8·15-s + 16-s − 4·20-s − 8·23-s + 6·25-s − 4·27-s − 8·33-s − 3·36-s + 16·37-s − 4·44-s + 12·45-s + 4·47-s − 2·48-s + 5·49-s − 8·53-s + 16·55-s − 6·59-s + 8·60-s − 64-s + 10·67-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.78·5-s + 9-s + 1.20·11-s + 0.577·12-s − 2.06·15-s + 1/4·16-s − 0.894·20-s − 1.66·23-s + 6/5·25-s − 0.769·27-s − 1.39·33-s − 1/2·36-s + 2.63·37-s − 0.603·44-s + 1.78·45-s + 0.583·47-s − 0.288·48-s + 5/7·49-s − 1.09·53-s + 2.15·55-s − 0.781·59-s + 1.03·60-s − 1/8·64-s + 1.22·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179182584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179182584\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 133 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−7.31747525484386711779156572191, −6.72287017439873493021694337978, −6.31454171215534259768821353127, −6.19788277398408073207241602551, −5.72242563720850320450028196836, −5.63634546852323474646941917415, −4.89710470451780339995352329200, −4.58866900743405234339974046107, −4.08658598910648355914945123242, −3.76053237473530730469100582845, −2.96044031243555198559811794522, −2.23390687859720587741936634015, −1.88016455723613903531822994736, −1.24563578358112386873387698380, −0.62537210499861875528573098859,
0.62537210499861875528573098859, 1.24563578358112386873387698380, 1.88016455723613903531822994736, 2.23390687859720587741936634015, 2.96044031243555198559811794522, 3.76053237473530730469100582845, 4.08658598910648355914945123242, 4.58866900743405234339974046107, 4.89710470451780339995352329200, 5.63634546852323474646941917415, 5.72242563720850320450028196836, 6.19788277398408073207241602551, 6.31454171215534259768821353127, 6.72287017439873493021694337978, 7.31747525484386711779156572191
