L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 7·7-s + 8·8-s + 9·9-s + 10·10-s − 14·14-s + 16·16-s + 6·17-s + 18·18-s − 18·19-s + 20·20-s + 25·25-s − 28·28-s + 32·32-s + 12·34-s − 35·35-s + 36·36-s − 66·37-s − 36·38-s + 40·40-s − 54·43-s + 45·45-s + 66·47-s + 49·49-s + 50·50-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 9-s + 10-s − 14-s + 16-s + 6/17·17-s + 18-s − 0.947·19-s + 20-s + 25-s − 28-s + 32-s + 6/17·34-s − 35-s + 36-s − 1.78·37-s − 0.947·38-s + 40-s − 1.25·43-s + 45-s + 1.40·47-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.386272435\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.386272435\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 6 T + p^{2} T^{2} \) |
| 19 | \( 1 + 18 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 66 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 54 T + p^{2} T^{2} \) |
| 47 | \( 1 - 66 T + p^{2} T^{2} \) |
| 53 | \( 1 + 34 T + p^{2} T^{2} \) |
| 59 | \( 1 - 62 T + p^{2} T^{2} \) |
| 61 | \( 1 + 102 T + p^{2} T^{2} \) |
| 67 | \( 1 + 6 T + p^{2} T^{2} \) |
| 71 | \( 1 + 138 T + p^{2} T^{2} \) |
| 73 | \( 1 + 106 T + p^{2} T^{2} \) |
| 79 | \( 1 + 122 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 166 T + p^{2} T^{2} \) |
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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
−11.98753079052968282722375304142, −10.46455080505231632821034533902, −10.14665921694186576197380035727, −8.906759835173856218868043347818, −7.26432319750209972519013061447, −6.51386300303661422448108744157, −5.62611870159730368448908227376, −4.39642458233515133871584042957, −3.12752722436231358419940060299, −1.74891837706868605034125083009,
1.74891837706868605034125083009, 3.12752722436231358419940060299, 4.39642458233515133871584042957, 5.62611870159730368448908227376, 6.51386300303661422448108744157, 7.26432319750209972519013061447, 8.906759835173856218868043347818, 10.14665921694186576197380035727, 10.46455080505231632821034533902, 11.98753079052968282722375304142