| L(s) = 1 | − 4-s − 4·7-s + 2·13-s − 3·16-s + 4·19-s + 4·28-s + 16·31-s + 14·37-s − 4·43-s − 2·49-s − 2·52-s − 14·61-s + 7·64-s + 20·67-s + 14·73-s − 4·76-s + 4·79-s − 8·91-s − 4·97-s − 16·103-s + 22·109-s + 12·112-s − 10·121-s − 16·124-s + 127-s + 131-s − 16·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s + 0.554·13-s − 3/4·16-s + 0.917·19-s + 0.755·28-s + 2.87·31-s + 2.30·37-s − 0.609·43-s − 2/7·49-s − 0.277·52-s − 1.79·61-s + 7/8·64-s + 2.44·67-s + 1.63·73-s − 0.458·76-s + 0.450·79-s − 0.838·91-s − 0.406·97-s − 1.57·103-s + 2.10·109-s + 1.13·112-s − 0.909·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611064296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611064296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.447821264943873333323012557906, −9.184223208127508094098709396938, −8.353485184807691348471645277877, −8.310513875915312226017998862339, −7.947618380449350930263817051252, −7.35589387037622311723691628928, −6.88594086988018500425242975268, −6.40823275604343529007871559736, −6.32232728731598886210263906668, −5.99549383601386101104243636214, −5.11771642284714285640371549606, −5.06272690183642288958815350402, −4.26475609025031521928762946341, −4.17243136874650212517370142112, −3.43432816631432105271551598640, −3.01046826617625559488310943354, −2.73028684377486285888744374386, −2.01476746180984598759487958200, −1.03192950689451170783495403320, −0.55671524723422210844814596870,
0.55671524723422210844814596870, 1.03192950689451170783495403320, 2.01476746180984598759487958200, 2.73028684377486285888744374386, 3.01046826617625559488310943354, 3.43432816631432105271551598640, 4.17243136874650212517370142112, 4.26475609025031521928762946341, 5.06272690183642288958815350402, 5.11771642284714285640371549606, 5.99549383601386101104243636214, 6.32232728731598886210263906668, 6.40823275604343529007871559736, 6.88594086988018500425242975268, 7.35589387037622311723691628928, 7.947618380449350930263817051252, 8.310513875915312226017998862339, 8.353485184807691348471645277877, 9.184223208127508094098709396938, 9.447821264943873333323012557906
