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 − 1.85·7-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 − 1.85·63-s − 0.194·67-s + 0.630·73-s + 1.79·79-s + 81-s + 23/7·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3935822416\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3935822416\) |
\(L(2)\) |
|
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
−9.577717426479420368779515890266, −9.158376545900250033237811636814, −7.51088355831290791095839984776, −7.02704569181704601844562747970, −6.19489569220104366113582384470, −5.47178115452758990275970262476, −4.44406817734147346073763160895, −3.41241412027901598510585152748, −2.21912643709397653505131714573, −0.36840013993712410346406489089,
0.36840013993712410346406489089, 2.21912643709397653505131714573, 3.41241412027901598510585152748, 4.44406817734147346073763160895, 5.47178115452758990275970262476, 6.19489569220104366113582384470, 7.02704569181704601844562747970, 7.51088355831290791095839984776, 9.158376545900250033237811636814, 9.577717426479420368779515890266