L(s) = 1 | − 2.43i·2-s + 5.19·3-s + 10.0·4-s + 25.9·5-s − 12.6i·6-s − 17.4·7-s − 63.4i·8-s + 27·9-s − 63.0i·10-s − 44.7i·11-s + 52.4·12-s − 157. i·13-s + 42.4i·14-s + 134.·15-s + 7.22·16-s − 328.·17-s + ⋯ |
L(s) = 1 | − 0.607i·2-s + 0.577·3-s + 0.630·4-s + 1.03·5-s − 0.350i·6-s − 0.356·7-s − 0.991i·8-s + 0.333·9-s − 0.630i·10-s − 0.369i·11-s + 0.364·12-s − 0.931i·13-s + 0.216i·14-s + 0.598·15-s + 0.0282·16-s − 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.182550973\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.182550973\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (1.10e3 + 3.30e3i)T \) |
good | 2 | \( 1 + 2.43iT - 16T^{2} \) |
| 5 | \( 1 - 25.9T + 625T^{2} \) |
| 7 | \( 1 + 17.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 44.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 157. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 328.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 678.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 424. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 870.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.38e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.56e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 503.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.00e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.42e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.52e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 5.70e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.79e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.71e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.66e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.45e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.91e3iT - 8.85e7T^{2} \) |
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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
−11.78986436278327670708616686515, −10.69835545975012440802966801596, −9.891798409196826283995386374751, −9.101826324900264228613504701347, −7.64541899078207669157688431745, −6.54881671613184720166545284655, −5.40676337747262340564384267769, −3.44339031969254939783032019802, −2.53216488356384332303225566456, −1.20250694368287816971032617320,
1.76477592992538165664031337366, 2.83769568787979979750792207660, 4.74871916196181677585677800640, 6.15321266553437875967873217580, 6.84507409144931327768848801005, 7.985334800865719286434114231165, 9.233614549206650673930198634530, 9.934506142736265513966605301297, 11.21598326505602273207121635101, 12.24709573290938060011630660254