| L(s) = 1 | − 2.05·2-s − 0.939·3-s + 2.21·4-s − 1.53·5-s + 1.92·6-s − 2.99·7-s − 0.435·8-s − 2.11·9-s + 3.15·10-s − 2.07·12-s − 7.07·13-s + 6.14·14-s + 1.44·15-s − 3.53·16-s + 1.84·17-s + 4.34·18-s − 5.42·19-s − 3.39·20-s + 2.81·21-s + 2.95·23-s + 0.409·24-s − 2.64·25-s + 14.5·26-s + 4.80·27-s − 6.62·28-s − 3.67·29-s − 2.96·30-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.542·3-s + 1.10·4-s − 0.686·5-s + 0.787·6-s − 1.13·7-s − 0.153·8-s − 0.705·9-s + 0.996·10-s − 0.599·12-s − 1.96·13-s + 1.64·14-s + 0.372·15-s − 0.882·16-s + 0.448·17-s + 1.02·18-s − 1.24·19-s − 0.759·20-s + 0.613·21-s + 0.616·23-s + 0.0835·24-s − 0.528·25-s + 2.84·26-s + 0.925·27-s − 1.25·28-s − 0.682·29-s − 0.540·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 + 0.939T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.99T + 7T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 0.747T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 - 7.02T + 79T^{2} \) |
| 83 | \( 1 + 4.22T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 1.06T + 97T^{2} \) |
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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
−7.56957917477543907903674224469, −7.10006449932702291563448804353, −6.44467937278672606109439463538, −5.60971380205144540012818765534, −4.77080701958720736606222420076, −3.90061659584322767247384908444, −2.82532833390021141364947243162, −2.17656758286798118111613580121, −0.62083855126025169404193658329, 0,
0.62083855126025169404193658329, 2.17656758286798118111613580121, 2.82532833390021141364947243162, 3.90061659584322767247384908444, 4.77080701958720736606222420076, 5.60971380205144540012818765534, 6.44467937278672606109439463538, 7.10006449932702291563448804353, 7.56957917477543907903674224469
