L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s − 6·17-s − 23-s + 25-s + 27-s + 6·29-s − 6·37-s − 2·39-s − 6·41-s − 45-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s − 12·59-s + 10·61-s + 2·65-s − 69-s − 6·73-s + 75-s + 8·79-s + 81-s + 6·85-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.248·65-s − 0.120·69-s − 0.702·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.405062090501522425432625806457, −7.83315783909893742198109494470, −6.87953484753163495120463930655, −6.42745323646683849474829733946, −5.06850952503014453195679085382, −4.50421650696143290281208242635, −3.54023389135049481483602881039, −2.67869569929630113057820760890, −1.67449794423814169224104342467, 0,
1.67449794423814169224104342467, 2.67869569929630113057820760890, 3.54023389135049481483602881039, 4.50421650696143290281208242635, 5.06850952503014453195679085382, 6.42745323646683849474829733946, 6.87953484753163495120463930655, 7.83315783909893742198109494470, 8.405062090501522425432625806457