L(s) = 1 | − 2.67·2-s + 2.89·3-s + 3.15·4-s − 3.34i·5-s − 7.74·6-s + (−4.11 + 5.65i)7-s + 2.26·8-s − 0.605·9-s + 8.94i·10-s − 15.3i·11-s + 9.13·12-s − 7.80·13-s + (11.0 − 15.1i)14-s − 9.68i·15-s − 18.6·16-s − 8.48·17-s + ⋯ |
L(s) = 1 | − 1.33·2-s + 0.965·3-s + 0.787·4-s − 0.668i·5-s − 1.29·6-s + (−0.588 + 0.808i)7-s + 0.283·8-s − 0.0672·9-s + 0.894i·10-s − 1.39i·11-s + 0.761·12-s − 0.600·13-s + (0.786 − 1.08i)14-s − 0.645i·15-s − 1.16·16-s − 0.499·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.534i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.106994 - 0.369070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.106994 - 0.369070i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (4.11 - 5.65i)T \) |
| 41 | \( 1 + (38.1 + 15.1i)T \) |
good | 2 | \( 1 + 2.67T + 4T^{2} \) |
| 3 | \( 1 - 2.89T + 9T^{2} \) |
| 5 | \( 1 + 3.34iT - 25T^{2} \) |
| 11 | \( 1 + 15.3iT - 121T^{2} \) |
| 13 | \( 1 + 7.80T + 169T^{2} \) |
| 17 | \( 1 + 8.48T + 289T^{2} \) |
| 19 | \( 1 + 0.826T + 361T^{2} \) |
| 23 | \( 1 + 2.74T + 529T^{2} \) |
| 29 | \( 1 - 15.8iT - 841T^{2} \) |
| 31 | \( 1 + 52.1iT - 961T^{2} \) |
| 37 | \( 1 + 39.7T + 1.36e3T^{2} \) |
| 43 | \( 1 + 71.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 0.993T + 2.20e3T^{2} \) |
| 53 | \( 1 - 10.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 59.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.55iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 49.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 171.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 41.4T + 9.40e3T^{2} \) |
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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
−10.98564922842053411794641580515, −9.808228385817754286843406217855, −9.034877882053886074531898995354, −8.609550413880150200435396701874, −7.912909266540294726985987794059, −6.52814251346326576051116588960, −5.17018464108266500306712248742, −3.38538684073497158123477939407, −2.11346350569432105632717204910, −0.24672473738149675538382616399,
1.90385133757200241360289294318, 3.17920977249955730092665278070, 4.63427706174880008772259076983, 6.86272202209769765946268481668, 7.20640185054433145336850993724, 8.300313207475055570948313796699, 9.148086005908635936964390929973, 10.08834211780429188370412621988, 10.39882836811437321639917061290, 11.72417190660628637399903659877