L(s) = 1 | − 5.85i·3-s + 5.78·5-s − 2.64i·7-s − 25.2·9-s + 3.01i·11-s − 9.78·13-s − 33.8i·15-s − 11.6·17-s − 25.5i·19-s − 15.4·21-s − 26.1i·23-s + 8.42·25-s + 95.4i·27-s − 1.56·29-s + 12.0i·31-s + ⋯ |
L(s) = 1 | − 1.95i·3-s + 1.15·5-s − 0.377i·7-s − 2.81·9-s + 0.274i·11-s − 0.752·13-s − 2.25i·15-s − 0.682·17-s − 1.34i·19-s − 0.737·21-s − 1.13i·23-s + 0.337·25-s + 3.53i·27-s − 0.0539·29-s + 0.387i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.492824882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.492824882\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 5.85iT - 9T^{2} \) |
| 5 | \( 1 - 5.78T + 25T^{2} \) |
| 11 | \( 1 - 3.01iT - 121T^{2} \) |
| 13 | \( 1 + 9.78T + 169T^{2} \) |
| 17 | \( 1 + 11.6T + 289T^{2} \) |
| 19 | \( 1 + 25.5iT - 361T^{2} \) |
| 23 | \( 1 + 26.1iT - 529T^{2} \) |
| 29 | \( 1 + 1.56T + 841T^{2} \) |
| 31 | \( 1 - 12.0iT - 961T^{2} \) |
| 37 | \( 1 - 70.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 73.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 12.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 7.75T + 9.40e3T^{2} \) |
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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
−10.56084113647791870387164281836, −9.347679794319261008993596861296, −8.517181640021359612134962174937, −7.39811898886323447772953359982, −6.76512465728026272920718297304, −6.04311885207005861881855625548, −4.89918291622010091872980034104, −2.66823486594741691264045967289, −1.99148467226165758722971871373, −0.58198020875608273440697108886,
2.31399975609983322602482682963, 3.50659355779609984779517548744, 4.64866518115680234741300589514, 5.55138267746605220854919383300, 6.15018473293076196846967225014, 8.026015273577628844150704517047, 9.044603543246757212015683027379, 9.753972514657824716870026043440, 10.07514510811921799230936170599, 11.13101308254436398207267098980