L(s) = 1 | − 2-s − 1.80·3-s + 4-s + 1.80·6-s − 1.55·7-s − 8-s + 0.246·9-s + 4.49·11-s − 1.80·12-s + 1.55·14-s + 16-s − 0.396·17-s − 0.246·18-s + 5.60·19-s + 2.80·21-s − 4.49·22-s + 3.02·23-s + 1.80·24-s + 4.96·27-s − 1.55·28-s − 0.692·29-s − 6.59·31-s − 32-s − 8.09·33-s + 0.396·34-s + 0.246·36-s − 8.98·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.735·6-s − 0.587·7-s − 0.353·8-s + 0.0823·9-s + 1.35·11-s − 0.520·12-s + 0.415·14-s + 0.250·16-s − 0.0960·17-s − 0.0582·18-s + 1.28·19-s + 0.611·21-s − 0.958·22-s + 0.631·23-s + 0.367·24-s + 0.954·27-s − 0.293·28-s − 0.128·29-s − 1.18·31-s − 0.176·32-s − 1.40·33-s + 0.0679·34-s + 0.0411·36-s − 1.47·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 17 | \( 1 + 0.396T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 + 0.692T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 + 8.64T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.20363806346287051443760628570, −6.72603794088675524044972166436, −6.31238119614408187286287266538, −5.38596293692668582750071792756, −4.96718115397100393994767249677, −3.63900489672802512318421220203, −3.23348364405094769535997179108, −1.87018751949396259551975712850, −1.01762206673394631168629824047, 0,
1.01762206673394631168629824047, 1.87018751949396259551975712850, 3.23348364405094769535997179108, 3.63900489672802512318421220203, 4.96718115397100393994767249677, 5.38596293692668582750071792756, 6.31238119614408187286287266538, 6.72603794088675524044972166436, 7.20363806346287051443760628570