L(s) = 1 | − 2.34i·2-s + (0.822 + 1.52i)3-s − 3.50·4-s + (0.5 + 0.866i)5-s + (3.57 − 1.92i)6-s + (2.00 + 1.73i)7-s + 3.53i·8-s + (−1.64 + 2.50i)9-s + (2.03 − 1.17i)10-s + (4.81 + 2.77i)11-s + (−2.88 − 5.34i)12-s + (−0.0395 − 0.0228i)13-s + (4.06 − 4.69i)14-s + (−0.909 + 1.47i)15-s + 1.27·16-s + (−0.320 − 0.554i)17-s + ⋯ |
L(s) = 1 | − 1.65i·2-s + (0.474 + 0.880i)3-s − 1.75·4-s + (0.223 + 0.387i)5-s + (1.46 − 0.787i)6-s + (0.756 + 0.654i)7-s + 1.24i·8-s + (−0.549 + 0.835i)9-s + (0.642 − 0.370i)10-s + (1.45 + 0.837i)11-s + (−0.831 − 1.54i)12-s + (−0.0109 − 0.00632i)13-s + (1.08 − 1.25i)14-s + (−0.234 + 0.380i)15-s + 0.319·16-s + (−0.0776 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46651 - 0.554974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46651 - 0.554974i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.822 - 1.52i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.00 - 1.73i)T \) |
good | 2 | \( 1 + 2.34iT - 2T^{2} \) |
| 11 | \( 1 + (-4.81 - 2.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0395 + 0.0228i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.320 + 0.554i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.73 + 3.31i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.23 + 3.02i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.95 + 3.44i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.48iT - 31T^{2} \) |
| 37 | \( 1 + (3.67 - 6.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.580 + 1.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.08 + 3.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 + 6.68iT - 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 7.67iT - 71T^{2} \) |
| 73 | \( 1 + (3.76 - 2.17i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + (0.725 + 1.25i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.24 + 16.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.49 - 3.17i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.40140058577463969286639869999, −10.73519383541330560325980404824, −9.807417503492046000179465683633, −9.120072162967858938398298586795, −8.397770520097214199435183336448, −6.60515735486917264952719392432, −4.82205059838901133690707560995, −4.21020161720309535271544369753, −2.85095822063703388223044238345, −1.90796222271734005967212710166,
1.35250017543434240251532144975, 3.77137082179797957167275771570, 5.05547359644966268316271109243, 6.31422711569956925438119801222, 6.81462165079297888593993643266, 7.915064142217505852913365166692, 8.605943731715949494340550510443, 9.181010738566086586256045271725, 10.85455908770058229341460147497, 12.02248721898896014571334108936