L(s) = 1 | + 0.414·2-s + (−0.707 + 1.22i)3-s − 1.82·4-s + (0.914 − 1.58i)5-s + (−0.292 + 0.507i)6-s − 1.58·8-s + (0.500 + 0.866i)9-s + (0.378 − 0.655i)10-s + (−0.292 + 0.507i)11-s + (1.29 − 2.23i)12-s + (−3.5 + 0.866i)13-s + (1.29 + 2.23i)15-s + 3·16-s − 5.82·17-s + (0.207 + 0.358i)18-s + (−3 − 5.19i)19-s + ⋯ |
L(s) = 1 | + 0.292·2-s + (−0.408 + 0.707i)3-s − 0.914·4-s + (0.408 − 0.708i)5-s + (−0.119 + 0.207i)6-s − 0.560·8-s + (0.166 + 0.288i)9-s + (0.119 − 0.207i)10-s + (−0.0883 + 0.152i)11-s + (0.373 − 0.646i)12-s + (−0.970 + 0.240i)13-s + (0.333 + 0.578i)15-s + 0.750·16-s − 1.41·17-s + (0.0488 + 0.0845i)18-s + (−0.688 − 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.5 - 0.866i)T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.914 + 1.58i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.292 - 0.507i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + (2.08 + 3.61i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.29 + 2.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 + (0.0857 + 0.148i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.70 - 2.95i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.82 + 3.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + (2.08 + 3.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.12 - 3.67i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.82 + 10.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.32 - 9.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.878 - 1.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + (-5.41 + 9.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.11612369182569308954140469769, −9.253008325006451091127163862066, −8.871708047891928019944508044385, −7.61228671327449205766684590908, −6.36846092663247169429493261788, −5.08562042895818766304510628573, −4.84823248232045327911259167677, −3.95720729443932931772682316162, −2.19592306652031372783635833388, 0,
1.95276472145636761828754934479, 3.39633099732029916415493261932, 4.53513858379358614263131347168, 5.63515935094153939837657519106, 6.45149239711639396395104514240, 7.21917998714636082607153865935, 8.352945097744837708475055744532, 9.255434022224386753122188716973, 10.16520675246298074068853173738