| L(s) = 1 | − 2-s − 1.22·3-s + 4-s + 4.12·5-s + 1.22·6-s − 0.673·7-s − 8-s − 1.50·9-s − 4.12·10-s − 11-s − 1.22·12-s − 1.15·13-s + 0.673·14-s − 5.04·15-s + 16-s − 5.09·17-s + 1.50·18-s − 6.95·19-s + 4.12·20-s + 0.823·21-s + 22-s + 4.97·23-s + 1.22·24-s + 12.0·25-s + 1.15·26-s + 5.50·27-s − 0.673·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.705·3-s + 0.5·4-s + 1.84·5-s + 0.498·6-s − 0.254·7-s − 0.353·8-s − 0.501·9-s − 1.30·10-s − 0.301·11-s − 0.352·12-s − 0.319·13-s + 0.179·14-s − 1.30·15-s + 0.250·16-s − 1.23·17-s + 0.354·18-s − 1.59·19-s + 0.922·20-s + 0.179·21-s + 0.213·22-s + 1.03·23-s + 0.249·24-s + 2.40·25-s + 0.225·26-s + 1.05·27-s − 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 - 4.12T + 5T^{2} \) |
| 7 | \( 1 + 0.673T + 7T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 6.95T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 8.14T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 + 6.43T + 67T^{2} \) |
| 71 | \( 1 + 9.86T + 71T^{2} \) |
| 73 | \( 1 + 8.78T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 + 19.3T + 97T^{2} \) |
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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
−8.431999167792044873270873304744, −6.87694490960395534974314285022, −6.56563404851703795341273121209, −6.02232665280913604297899351372, −5.20852446859681022482610178986, −4.56475680510707174520907341945, −2.82421387963835722082630784324, −2.39721397604632208543966117370, −1.33338113960605940530826590403, 0,
1.33338113960605940530826590403, 2.39721397604632208543966117370, 2.82421387963835722082630784324, 4.56475680510707174520907341945, 5.20852446859681022482610178986, 6.02232665280913604297899351372, 6.56563404851703795341273121209, 6.87694490960395534974314285022, 8.431999167792044873270873304744
