L(s) = 1 | − 3-s + 5-s + (1.26 + 2.19i)7-s + 9-s + (0.0636 + 0.110i)11-s + (−2.22 + 3.85i)13-s − 15-s + (0.129 − 0.224i)17-s + (3.21 − 5.56i)19-s + (−1.26 − 2.19i)21-s + (3.09 − 5.35i)23-s + 25-s − 27-s + (−0.920 − 1.59i)29-s + (−2.61 − 4.53i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (0.477 + 0.827i)7-s + 0.333·9-s + (0.0191 + 0.0332i)11-s + (−0.617 + 1.06i)13-s − 0.258·15-s + (0.0314 − 0.0544i)17-s + (0.737 − 1.27i)19-s + (−0.275 − 0.477i)21-s + (0.644 − 1.11i)23-s + 0.200·25-s − 0.192·27-s + (−0.170 − 0.295i)29-s + (−0.470 − 0.814i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.795566728\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795566728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-7.76 - 2.58i)T \) |
good | 7 | \( 1 + (-1.26 - 2.19i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0636 - 0.110i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.22 - 3.85i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.129 + 0.224i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.21 + 5.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.920 + 1.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.61 + 4.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.92 + 8.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.579 - 1.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 2.35T + 43T^{2} \) |
| 47 | \( 1 + (-5.48 - 9.50i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 0.400T + 59T^{2} \) |
| 61 | \( 1 + (-2.78 + 4.82i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.62 - 6.28i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.00 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.753 - 1.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.21 - 5.56i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 1.27T + 89T^{2} \) |
| 97 | \( 1 + (2.93 - 5.07i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.586308976081480817261454847355, −7.57685666143692469744246793901, −6.95039009487723144468199974959, −6.19412230379722805575507074288, −5.44496476145265193277606958354, −4.82384847499998091366796477935, −4.11236819556748116156718700219, −2.64167839352266048280832092520, −2.12594034638440251200920064983, −0.77069607701553889850816002746,
0.862726364644656957896429755955, 1.69416362419000046442642041230, 3.04658105307728561637809726095, 3.82860448744456961763419225943, 4.90537921717008944545923058560, 5.39789081791328154059068532794, 6.08841712735860375576260038965, 7.13582830800905872009550582626, 7.54467460085873622295472294262, 8.284301497392635546033634636980