| L(s) = 1 | − 3·5-s + 7-s − 3·11-s + 4·13-s − 6·17-s + 2·19-s − 6·23-s + 5·25-s + 18·29-s − 7·31-s − 3·35-s + 10·37-s + 8·43-s + 12·47-s − 6·49-s + 3·53-s + 9·55-s − 3·59-s + 4·61-s − 12·65-s + 2·67-s − 2·73-s − 3·77-s + 5·79-s − 18·83-s + 18·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.904·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 25-s + 3.34·29-s − 1.25·31-s − 0.507·35-s + 1.64·37-s + 1.21·43-s + 1.75·47-s − 6/7·49-s + 0.412·53-s + 1.21·55-s − 0.390·59-s + 0.512·61-s − 1.48·65-s + 0.244·67-s − 0.234·73-s − 0.341·77-s + 0.562·79-s − 1.97·83-s + 1.95·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267996609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267996609\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−10.10808856957645975351917607242, −9.975646504346794274877075024258, −9.050090566051939734338331164876, −8.970626824856214767523167828131, −8.373558269339540129128116687511, −8.044268050005006800554241031652, −7.87791013055404234309224584781, −7.35038052014407444465084794476, −6.65014892356314452493807033363, −6.62070536059120507158464048755, −5.76847484041962998092619467125, −5.55560373656079630959494587094, −4.73090435828397489172979158510, −4.28473550331234860402312293648, −4.18710654418748340735998194751, −3.50091624768140207978449546864, −2.66931115540000292494786191131, −2.52633180207511502028265536695, −1.37683711622456274284589108222, −0.55254619880897760023605978024,
0.55254619880897760023605978024, 1.37683711622456274284589108222, 2.52633180207511502028265536695, 2.66931115540000292494786191131, 3.50091624768140207978449546864, 4.18710654418748340735998194751, 4.28473550331234860402312293648, 4.73090435828397489172979158510, 5.55560373656079630959494587094, 5.76847484041962998092619467125, 6.62070536059120507158464048755, 6.65014892356314452493807033363, 7.35038052014407444465084794476, 7.87791013055404234309224584781, 8.044268050005006800554241031652, 8.373558269339540129128116687511, 8.970626824856214767523167828131, 9.050090566051939734338331164876, 9.975646504346794274877075024258, 10.10808856957645975351917607242
