L(s) = 1 | − 3-s − 1.41·7-s + 9-s − 1.41·13-s − 6·17-s + 6·19-s + 1.41·21-s + 8.48·23-s − 5·25-s − 27-s + 8.48·29-s + 1.41·31-s − 7.07·37-s + 1.41·39-s − 6·41-s + 6·43-s − 8.48·47-s − 5·49-s + 6·51-s − 8.48·53-s − 6·57-s + 7.07·61-s − 1.41·63-s − 4·67-s − 8.48·69-s − 8.48·71-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.534·7-s + 0.333·9-s − 0.392·13-s − 1.45·17-s + 1.37·19-s + 0.308·21-s + 1.76·23-s − 25-s − 0.192·27-s + 1.57·29-s + 0.254·31-s − 1.16·37-s + 0.226·39-s − 0.937·41-s + 0.914·43-s − 1.23·47-s − 0.714·49-s + 0.840·51-s − 1.16·53-s − 0.794·57-s + 0.905·61-s − 0.178·63-s − 0.488·67-s − 1.02·69-s − 1.00·71-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.398184701957722667865070661577, −7.36995787698778870278073785273, −6.79212508856785945628080630693, −6.17569000321786338184132194099, −5.11431987678555296322203898172, −4.67944689414478614821404211321, −3.48666376572590897771801051832, −2.66775304250655710071961612304, −1.34412228164732750790819644302, 0,
1.34412228164732750790819644302, 2.66775304250655710071961612304, 3.48666376572590897771801051832, 4.67944689414478614821404211321, 5.11431987678555296322203898172, 6.17569000321786338184132194099, 6.79212508856785945628080630693, 7.36995787698778870278073785273, 8.398184701957722667865070661577