L(s) = 1 | − 2-s + 4-s − 8-s − 11-s + 3·13-s + 16-s + 4·17-s − 19-s + 22-s − 3·23-s − 3·26-s − 5·29-s − 3·31-s − 32-s − 4·34-s − 12·37-s + 38-s − 8·41-s − 5·43-s − 44-s + 3·46-s + 8·47-s − 7·49-s + 3·52-s + 10·53-s + 5·58-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.301·11-s + 0.832·13-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.213·22-s − 0.625·23-s − 0.588·26-s − 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.685·34-s − 1.97·37-s + 0.162·38-s − 1.24·41-s − 0.762·43-s − 0.150·44-s + 0.442·46-s + 1.16·47-s − 49-s + 0.416·52-s + 1.37·53-s + 0.656·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.118572909483364600781690126383, −7.19779608745618137850854055529, −6.65761257701084225626651687217, −5.67817783393391267816171313305, −5.22902313705319593089010039903, −3.88364560980300477723210369750, −3.35060807565463995691259956981, −2.16693507589082053541431997990, −1.33599545543627131176586749757, 0,
1.33599545543627131176586749757, 2.16693507589082053541431997990, 3.35060807565463995691259956981, 3.88364560980300477723210369750, 5.22902313705319593089010039903, 5.67817783393391267816171313305, 6.65761257701084225626651687217, 7.19779608745618137850854055529, 8.118572909483364600781690126383