L(s) = 1 | − 2·5-s − 7-s − 4·13-s − 6·17-s − 2·19-s + 2·23-s − 25-s − 2·29-s + 8·31-s + 2·35-s − 2·37-s + 6·41-s + 2·43-s − 10·47-s + 49-s − 14·53-s − 12·61-s + 8·65-s − 8·67-s + 6·71-s − 6·73-s − 4·79-s − 4·83-s + 12·85-s − 12·89-s + 4·91-s + 4·95-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 0.417·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.937·41-s + 0.304·43-s − 1.45·47-s + 1/7·49-s − 1.92·53-s − 1.53·61-s + 0.992·65-s − 0.977·67-s + 0.712·71-s − 0.702·73-s − 0.450·79-s − 0.439·83-s + 1.30·85-s − 1.27·89-s + 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.89607323472669, −14.34710202454656, −13.78649099581294, −13.19322495665838, −12.74138582955489, −12.31655684293554, −11.72036444024872, −11.31366232814083, −10.82475151001673, −10.23016380726225, −9.632369496271067, −9.176881859615861, −8.622783116895273, −7.968152881606089, −7.611304463338413, −6.971521112101192, −6.471348883562888, −5.993616802571026, −5.063839156389650, −4.505769368729288, −4.266751476575319, −3.366113107219481, −2.808195359021621, −2.196554925183986, −1.310484621284780, 0, 0,
1.310484621284780, 2.196554925183986, 2.808195359021621, 3.366113107219481, 4.266751476575319, 4.505769368729288, 5.063839156389650, 5.993616802571026, 6.471348883562888, 6.971521112101192, 7.611304463338413, 7.968152881606089, 8.622783116895273, 9.176881859615861, 9.632369496271067, 10.23016380726225, 10.82475151001673, 11.31366232814083, 11.72036444024872, 12.31655684293554, 12.74138582955489, 13.19322495665838, 13.78649099581294, 14.34710202454656, 14.89607323472669