| L(s) = 1 | + 5-s + 4·7-s − 2·11-s − 4·13-s + 2·17-s − 10·19-s − 5·23-s − 8·29-s − 7·31-s + 4·35-s − 12·37-s − 6·41-s + 2·43-s − 8·47-s + 7·49-s + 18·53-s − 2·55-s − 4·59-s − 13·61-s − 4·65-s + 10·67-s − 12·71-s − 12·73-s − 8·77-s − 9·79-s + 17·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.485·17-s − 2.29·19-s − 1.04·23-s − 1.48·29-s − 1.25·31-s + 0.676·35-s − 1.97·37-s − 0.937·41-s + 0.304·43-s − 1.16·47-s + 49-s + 2.47·53-s − 0.269·55-s − 0.520·59-s − 1.66·61-s − 0.496·65-s + 1.22·67-s − 1.42·71-s − 1.40·73-s − 0.911·77-s − 1.01·79-s + 1.86·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4187902429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4187902429\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 17 T + 206 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.904140330653818337165178648858, −8.367117944663208171186397838457, −8.216014257130467112720472354466, −7.66903149009137062757434750620, −7.36661213231094987205221661183, −7.17073448359366060728582154545, −6.63579768194865165219740901249, −6.04419303541784649137779314547, −5.83065703710561033975219988472, −5.23676594262831427400264063527, −5.20294727933758717150986698979, −4.51834540101091665565037881762, −4.41078073699718774615675132324, −3.68260359235524166952283971697, −3.44867414393349377751825320280, −2.53461070912300990198204588456, −2.21756807244551354036819310769, −1.73362831597264875195716787381, −1.59253611192481790698860195029, −0.17931063121907144766927808265,
0.17931063121907144766927808265, 1.59253611192481790698860195029, 1.73362831597264875195716787381, 2.21756807244551354036819310769, 2.53461070912300990198204588456, 3.44867414393349377751825320280, 3.68260359235524166952283971697, 4.41078073699718774615675132324, 4.51834540101091665565037881762, 5.20294727933758717150986698979, 5.23676594262831427400264063527, 5.83065703710561033975219988472, 6.04419303541784649137779314547, 6.63579768194865165219740901249, 7.17073448359366060728582154545, 7.36661213231094987205221661183, 7.66903149009137062757434750620, 8.216014257130467112720472354466, 8.367117944663208171186397838457, 8.904140330653818337165178648858
