L(s) = 1 | − 2-s − 2.24·3-s + 4-s − 4.22·5-s + 2.24·6-s − 3.19·7-s − 8-s + 2.01·9-s + 4.22·10-s + 4.83·11-s − 2.24·12-s + 1.15·13-s + 3.19·14-s + 9.46·15-s + 16-s − 0.359·17-s − 2.01·18-s − 4.81·19-s − 4.22·20-s + 7.15·21-s − 4.83·22-s + 3.06·23-s + 2.24·24-s + 12.8·25-s − 1.15·26-s + 2.19·27-s − 3.19·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 1.88·5-s + 0.914·6-s − 1.20·7-s − 0.353·8-s + 0.673·9-s + 1.33·10-s + 1.45·11-s − 0.646·12-s + 0.321·13-s + 0.853·14-s + 2.44·15-s + 0.250·16-s − 0.0872·17-s − 0.476·18-s − 1.10·19-s − 0.944·20-s + 1.56·21-s − 1.03·22-s + 0.639·23-s + 0.457·24-s + 2.57·25-s − 0.227·26-s + 0.422·27-s − 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 0.359T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 0.874T + 37T^{2} \) |
| 41 | \( 1 + 8.79T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 + 0.0537T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + 7.13T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 9.83T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.848066794977402684095861131306, −8.498201913851313695543392587101, −7.26386740276714691956976911556, −6.60415571299171051593120279729, −6.28267130379128128214458443360, −4.80118800519933925731681664516, −3.94586261962133230686912568941, −3.11295197640407919981755569129, −0.967422723558437876828045005312, 0,
0.967422723558437876828045005312, 3.11295197640407919981755569129, 3.94586261962133230686912568941, 4.80118800519933925731681664516, 6.28267130379128128214458443360, 6.60415571299171051593120279729, 7.26386740276714691956976911556, 8.498201913851313695543392587101, 8.848066794977402684095861131306