| L(s) = 1 | − 1.73i·3-s − 2.23i·5-s + (6.69 − 2.03i)7-s − 2.99·9-s + 2.03·11-s − 18.0i·13-s − 3.87·15-s + 1.07i·17-s − 28.7i·19-s + (−3.52 − 11.6i)21-s − 24.8·23-s − 5.00·25-s + 5.19i·27-s − 38.4·29-s + 44.0i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + (0.956 − 0.290i)7-s − 0.333·9-s + 0.184·11-s − 1.38i·13-s − 0.258·15-s + 0.0631i·17-s − 1.51i·19-s + (−0.167 − 0.552i)21-s − 1.08·23-s − 0.200·25-s + 0.192i·27-s − 1.32·29-s + 1.42i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.531861836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531861836\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (-6.69 + 2.03i)T \) |
| good | 11 | \( 1 - 2.03T + 121T^{2} \) |
| 13 | \( 1 + 18.0iT - 169T^{2} \) |
| 17 | \( 1 - 1.07iT - 289T^{2} \) |
| 19 | \( 1 + 28.7iT - 361T^{2} \) |
| 23 | \( 1 + 24.8T + 529T^{2} \) |
| 29 | \( 1 + 38.4T + 841T^{2} \) |
| 31 | \( 1 - 44.0iT - 961T^{2} \) |
| 37 | \( 1 - 37.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.58T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 95.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 95.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 28.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 29.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 67.6iT - 9.40e3T^{2} \) |
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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
−8.619391610923054342176781698951, −7.987586411082721205167409004720, −7.39702048935393330850479670017, −6.43175356555352906139858829052, −5.42537424592423788258920917684, −4.84238450338056833950258508961, −3.71334537897210038396500845255, −2.52959738493888352595854171491, −1.42685219860972459842189600687, −0.40205013671143957531500145684,
1.61592428847380985862498426141, 2.47092172834042420222970187803, 4.05815526068457405788263085294, 4.17190126068205109405859239520, 5.62447131661508289550279532982, 6.04281518715411640871323090568, 7.31210650083444320710719812504, 7.909995288296403846609341211699, 8.850379379790131345089260896505, 9.498175423937905936064639658008
