| L(s) = 1 | + 3·3-s + 13·7-s + 9·9-s − 23·13-s + 11·19-s + 39·21-s + 27·27-s + 59·31-s − 26·37-s − 69·39-s − 83·43-s + 120·49-s + 33·57-s − 121·61-s + 117·63-s + 13·67-s + 46·73-s − 142·79-s + 81·81-s − 299·91-s + 177·93-s − 167·97-s − 194·103-s + 71·109-s − 78·111-s − 207·117-s + ⋯ |
| L(s) = 1 | + 3-s + 13/7·7-s + 9-s − 1.76·13-s + 0.578·19-s + 13/7·21-s + 27-s + 1.90·31-s − 0.702·37-s − 1.76·39-s − 1.93·43-s + 2.44·49-s + 0.578·57-s − 1.98·61-s + 13/7·63-s + 0.194·67-s + 0.630·73-s − 1.79·79-s + 81-s − 3.28·91-s + 1.90·93-s − 1.72·97-s − 1.88·103-s + 0.651·109-s − 0.702·111-s − 1.76·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.624844647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.624844647\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 59 T + p^{2} T^{2} \) |
| 37 | \( 1 + 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 83 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 - 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 167 T + p^{2} T^{2} \) |
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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
−11.67435859645297621403052964776, −10.43408980830569027442693636095, −9.619731914503858134084662166870, −8.460112652955810907226618264267, −7.84967749173925088632960273151, −7.01369605587529416804517469514, −5.11185734724104878503839759333, −4.45522778289589384118515576210, −2.77684077723795404072676856845, −1.60138498571055517302612665151,
1.60138498571055517302612665151, 2.77684077723795404072676856845, 4.45522778289589384118515576210, 5.11185734724104878503839759333, 7.01369605587529416804517469514, 7.84967749173925088632960273151, 8.460112652955810907226618264267, 9.619731914503858134084662166870, 10.43408980830569027442693636095, 11.67435859645297621403052964776
