L(s) = 1 | − 2·5-s − 6·7-s + 2·11-s − 6·13-s + 2·19-s + 12·23-s + 3·25-s + 10·29-s − 4·31-s + 12·35-s + 2·37-s − 6·41-s − 6·43-s + 20·47-s + 16·49-s − 4·55-s + 12·59-s + 4·61-s + 12·65-s − 16·67-s − 4·71-s − 12·73-s − 12·77-s − 16·79-s − 24·83-s − 2·89-s + 36·91-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 2.26·7-s + 0.603·11-s − 1.66·13-s + 0.458·19-s + 2.50·23-s + 3/5·25-s + 1.85·29-s − 0.718·31-s + 2.02·35-s + 0.328·37-s − 0.937·41-s − 0.914·43-s + 2.91·47-s + 16/7·49-s − 0.539·55-s + 1.56·59-s + 0.512·61-s + 1.48·65-s − 1.95·67-s − 0.474·71-s − 1.40·73-s − 1.36·77-s − 1.80·79-s − 2.63·83-s − 0.211·89-s + 3.77·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 104 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.41878028816641078108945960248, −7.31620255885216586085995738056, −7.08020834164659534722953366510, −6.99870648392699729960073562082, −6.42198922954966163514603293969, −6.23516895237627120478645743339, −5.50439014394300644246921611079, −5.48382255323977625133337996697, −4.88184237790997095228181877713, −4.50083331764340565026248984698, −4.15373916559067760898867014933, −3.77803383958968206510459517733, −3.17488554068611391567029422649, −2.99565040972473784492079679255, −2.76223583260354349877241079365, −2.35624204337399972023086189475, −1.23141172400915812027642228815, −1.03699444365904348453961048063, 0, 0,
1.03699444365904348453961048063, 1.23141172400915812027642228815, 2.35624204337399972023086189475, 2.76223583260354349877241079365, 2.99565040972473784492079679255, 3.17488554068611391567029422649, 3.77803383958968206510459517733, 4.15373916559067760898867014933, 4.50083331764340565026248984698, 4.88184237790997095228181877713, 5.48382255323977625133337996697, 5.50439014394300644246921611079, 6.23516895237627120478645743339, 6.42198922954966163514603293969, 6.99870648392699729960073562082, 7.08020834164659534722953366510, 7.31620255885216586085995738056, 7.41878028816641078108945960248