| L(s) = 1 | − 2·3-s − 6·7-s − 9-s + 6·13-s + 2·17-s + 2·19-s + 12·21-s − 10·23-s + 6·27-s − 6·29-s − 4·31-s − 12·39-s − 12·43-s + 15·49-s − 4·51-s − 6·53-s − 4·57-s + 6·59-s + 20·61-s + 6·63-s − 18·67-s + 20·69-s + 24·71-s + 6·73-s − 4·79-s − 4·81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s − 1/3·9-s + 1.66·13-s + 0.485·17-s + 0.458·19-s + 2.61·21-s − 2.08·23-s + 1.15·27-s − 1.11·29-s − 0.718·31-s − 1.92·39-s − 1.82·43-s + 15/7·49-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 2.56·61-s + 0.755·63-s − 2.19·67-s + 2.40·69-s + 2.84·71-s + 0.702·73-s − 0.450·79-s − 4/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6975143310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6975143310\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.057334228531402187935844275678, −7.80947523185228543754495769314, −7.02121660362294126842324014634, −6.89591318699788309308585209639, −6.60925611297014590718413892820, −6.25252444222672631133492188417, −5.90490562452928981178701799306, −5.78064395466502431290077905884, −5.36874951342787396190054201957, −5.22842547368811812480699610547, −4.44900296610593197984581244290, −3.88724104573993141454737070634, −3.67408930959227333450210990623, −3.55261147468150429076037296787, −2.93655084899245770171479323750, −2.69100625026331365960658791130, −1.82375861032489316295646955709, −1.58695534124159590387967947251, −0.56987093657706897724803345373, −0.38117839939102305150315056283,
0.38117839939102305150315056283, 0.56987093657706897724803345373, 1.58695534124159590387967947251, 1.82375861032489316295646955709, 2.69100625026331365960658791130, 2.93655084899245770171479323750, 3.55261147468150429076037296787, 3.67408930959227333450210990623, 3.88724104573993141454737070634, 4.44900296610593197984581244290, 5.22842547368811812480699610547, 5.36874951342787396190054201957, 5.78064395466502431290077905884, 5.90490562452928981178701799306, 6.25252444222672631133492188417, 6.60925611297014590718413892820, 6.89591318699788309308585209639, 7.02121660362294126842324014634, 7.80947523185228543754495769314, 8.057334228531402187935844275678
