| L(s) = 1 | + 2·5-s − 4·7-s − 2·11-s + 8·19-s + 8·23-s + 3·25-s − 4·29-s + 4·31-s − 8·35-s − 20·37-s − 4·41-s + 12·43-s + 8·47-s + 4·49-s + 4·53-s − 4·55-s + 4·59-s − 4·61-s + 8·67-s + 12·71-s − 16·73-s + 8·77-s + 16·79-s + 20·83-s + 12·89-s + 16·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.603·11-s + 1.83·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s + 0.718·31-s − 1.35·35-s − 3.28·37-s − 0.624·41-s + 1.82·43-s + 1.16·47-s + 4/7·49-s + 0.549·53-s − 0.539·55-s + 0.520·59-s − 0.512·61-s + 0.977·67-s + 1.42·71-s − 1.87·73-s + 0.911·77-s + 1.80·79-s + 2.19·83-s + 1.27·89-s + 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.858976627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.858976627\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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
−8.688753792895882553474355636060, −8.528633342332084164473615578680, −7.58450179255986974046041419750, −7.57984812212914586957889886155, −7.18001135667186017149315877847, −6.79868852388129303641608044586, −6.42608260458580955634514258103, −6.13329509021090842591414572928, −5.55468616977031845343059360524, −5.37036531021693198767950003250, −5.00324700012710475441794134109, −4.72103364291698952059877680068, −3.77159105777784424880462864900, −3.55887413332452762988172788514, −3.17822917291874172861441739335, −2.84032870176974380774474966247, −2.24106950928213074185701560452, −1.84488793571576616093021777299, −0.953949788459649734628350040479, −0.59123861440622527349243450724,
0.59123861440622527349243450724, 0.953949788459649734628350040479, 1.84488793571576616093021777299, 2.24106950928213074185701560452, 2.84032870176974380774474966247, 3.17822917291874172861441739335, 3.55887413332452762988172788514, 3.77159105777784424880462864900, 4.72103364291698952059877680068, 5.00324700012710475441794134109, 5.37036531021693198767950003250, 5.55468616977031845343059360524, 6.13329509021090842591414572928, 6.42608260458580955634514258103, 6.79868852388129303641608044586, 7.18001135667186017149315877847, 7.57984812212914586957889886155, 7.58450179255986974046041419750, 8.528633342332084164473615578680, 8.688753792895882553474355636060
