| L(s) = 1 | + 2·5-s + 4·11-s + 4·13-s − 2·17-s − 4·19-s − 8·23-s + 5·25-s − 12·29-s + 8·31-s − 6·37-s − 12·41-s + 8·43-s − 2·53-s + 8·55-s − 4·59-s − 2·61-s + 8·65-s + 4·67-s − 16·71-s + 10·73-s + 8·79-s − 8·83-s − 4·85-s + 6·89-s − 8·95-s − 4·97-s + 18·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 25-s − 2.22·29-s + 1.43·31-s − 0.986·37-s − 1.87·41-s + 1.21·43-s − 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.992·65-s + 0.488·67-s − 1.89·71-s + 1.17·73-s + 0.900·79-s − 0.878·83-s − 0.433·85-s + 0.635·89-s − 0.820·95-s − 0.406·97-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.806521443817236436082391013192, −8.523206547480919326328770445403, −8.058021560883418098093678656701, −7.71420777938584150005123227260, −7.22372983546029733968744825466, −6.65361833739716446526175153808, −6.49136920534852276201503134624, −6.29996019405270781317064729077, −5.65975145181356188248389810486, −5.64093376788795489009802645719, −4.97841640180916414662446311510, −4.42682838833444422738022997011, −4.13534507678828423403722059559, −3.73300437114305883411929180311, −3.33212284215746924064633671685, −2.75940405472030757182818077422, −1.94343588405441942659559806328, −1.89391777779346160987247761598, −1.36884072233065250071738913523, −0.47793955577332570900640088161,
0.47793955577332570900640088161, 1.36884072233065250071738913523, 1.89391777779346160987247761598, 1.94343588405441942659559806328, 2.75940405472030757182818077422, 3.33212284215746924064633671685, 3.73300437114305883411929180311, 4.13534507678828423403722059559, 4.42682838833444422738022997011, 4.97841640180916414662446311510, 5.64093376788795489009802645719, 5.65975145181356188248389810486, 6.29996019405270781317064729077, 6.49136920534852276201503134624, 6.65361833739716446526175153808, 7.22372983546029733968744825466, 7.71420777938584150005123227260, 8.058021560883418098093678656701, 8.523206547480919326328770445403, 8.806521443817236436082391013192
