L(s) = 1 | + 1.54·2-s + 3-s + 0.398·4-s + 1.02·5-s + 1.54·6-s − 2.48·8-s + 9-s + 1.58·10-s − 6.58·11-s + 0.398·12-s + 0.0967·13-s + 1.02·15-s − 4.63·16-s + 2.79·17-s + 1.54·18-s − 5.99·19-s + 0.408·20-s − 10.1·22-s + 23-s − 2.48·24-s − 3.94·25-s + 0.149·26-s + 27-s + 2.41·29-s + 1.58·30-s − 10.1·31-s − 2.22·32-s + ⋯ |
L(s) = 1 | + 1.09·2-s + 0.577·3-s + 0.199·4-s + 0.458·5-s + 0.632·6-s − 0.877·8-s + 0.333·9-s + 0.502·10-s − 1.98·11-s + 0.114·12-s + 0.0268·13-s + 0.264·15-s − 1.15·16-s + 0.677·17-s + 0.365·18-s − 1.37·19-s + 0.0913·20-s − 2.17·22-s + 0.208·23-s − 0.506·24-s − 0.789·25-s + 0.0293·26-s + 0.192·27-s + 0.447·29-s + 0.290·30-s − 1.81·31-s − 0.392·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 11 | \( 1 + 6.58T + 11T^{2} \) |
| 13 | \( 1 - 0.0967T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 0.972T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 1.63T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 0.921T + 97T^{2} \) |
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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
−8.112250219250831334825761216459, −7.59971131385375648967264640475, −6.52456757089302438950874527327, −5.70808007402360538255714998335, −5.20055495861619986182085116480, −4.39281101903202512743102028716, −3.50433591118086768680096856838, −2.71872806474941342819360105742, −1.99026214723177515932425598026, 0,
1.99026214723177515932425598026, 2.71872806474941342819360105742, 3.50433591118086768680096856838, 4.39281101903202512743102028716, 5.20055495861619986182085116480, 5.70808007402360538255714998335, 6.52456757089302438950874527327, 7.59971131385375648967264640475, 8.112250219250831334825761216459