L(s) = 1 | − 2-s + (1.48 + 0.894i)3-s + 4-s + (0.990 − 2.00i)5-s + (−1.48 − 0.894i)6-s + (−2.20 − 1.46i)7-s − 8-s + (1.40 + 2.65i)9-s + (−0.990 + 2.00i)10-s + (0.646 − 0.372i)11-s + (1.48 + 0.894i)12-s + (−1.67 − 2.89i)13-s + (2.20 + 1.46i)14-s + (3.26 − 2.08i)15-s + 16-s + (4.20 + 2.42i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.856 + 0.516i)3-s + 0.5·4-s + (0.442 − 0.896i)5-s + (−0.605 − 0.365i)6-s + (−0.832 − 0.554i)7-s − 0.353·8-s + (0.466 + 0.884i)9-s + (−0.313 + 0.633i)10-s + (0.194 − 0.112i)11-s + (0.428 + 0.258i)12-s + (−0.463 − 0.802i)13-s + (0.588 + 0.391i)14-s + (0.842 − 0.539i)15-s + 0.250·16-s + (1.01 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38723 - 0.435486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38723 - 0.435486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.48 - 0.894i)T \) |
| 5 | \( 1 + (-0.990 + 2.00i)T \) |
| 7 | \( 1 + (2.20 + 1.46i)T \) |
good | 11 | \( 1 + (-0.646 + 0.372i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.67 + 2.89i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.20 - 2.42i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.50 + 3.75i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.933 + 1.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.644 + 0.371i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.00iT - 31T^{2} \) |
| 37 | \( 1 + (-5.78 + 3.33i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.849 - 1.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.64 + 4.41i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (-2.94 + 5.09i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 - 1.71iT - 61T^{2} \) |
| 67 | \( 1 - 4.53iT - 67T^{2} \) |
| 71 | \( 1 - 4.17iT - 71T^{2} \) |
| 73 | \( 1 + (-5.10 + 8.84i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 + (-2.26 - 1.30i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.58 - 13.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.32 - 4.01i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.978806843890429553431986559192, −9.727261194519824148328314525619, −8.968903227887773165712406218308, −7.975875592732782220798230348116, −7.36931027101159289627729859687, −5.99264534470863508458321716755, −4.95528532660963028037642055328, −3.66540897247197022772014851716, −2.65202199322678509134503276112, −0.998496106905633467199322371680,
1.58764646984334426112194054823, 2.82002397047360762972777111439, 3.46149565202752002335748194343, 5.52939776356615071934997641307, 6.58657433970841389220583125543, 7.17805190186942553337256118078, 7.986614299519085826088669294263, 9.207241449717363876434148182455, 9.638087393956398011753279943211, 10.23237379192823205704879163690