L(s) = 1 | + (−1.21 − 2.10i)2-s + (0.376 + 0.652i)3-s + (−1.95 + 3.39i)4-s − 0.341·5-s + (0.916 − 1.58i)6-s + 4.65·8-s + (1.21 − 2.10i)9-s + (0.415 + 0.719i)10-s + (1.21 + 2.10i)11-s − 2.95·12-s + (−2.50 + 2.59i)13-s + (−0.128 − 0.222i)15-s + (−1.74 − 3.02i)16-s + (−0.974 + 1.68i)17-s − 5.91·18-s + (−3.14 + 5.44i)19-s + ⋯ |
L(s) = 1 | + (−0.859 − 1.48i)2-s + (0.217 + 0.376i)3-s + (−0.978 + 1.69i)4-s − 0.152·5-s + (0.374 − 0.647i)6-s + 1.64·8-s + (0.405 − 0.702i)9-s + (0.131 + 0.227i)10-s + (0.366 + 0.635i)11-s − 0.851·12-s + (−0.693 + 0.720i)13-s + (−0.0332 − 0.0575i)15-s + (−0.437 − 0.757i)16-s + (−0.236 + 0.409i)17-s − 1.39·18-s + (−0.721 + 1.24i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755216 + 0.00948345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755216 + 0.00948345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.50 - 2.59i)T \) |
good | 2 | \( 1 + (1.21 + 2.10i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.376 - 0.652i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.341T + 5T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.974 - 1.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.14 - 5.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.22 + 3.84i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.26 - 10.8i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.20 + 7.28i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 + (-0.313 + 0.542i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.571 - 0.990i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.74 - 8.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + (-6.22 - 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.13 + 8.90i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.35485341885716250528492233085, −9.755471450692216214522047065176, −9.293994723252685435559612367807, −8.332052993484195798280829631030, −7.40915010236327681443229541896, −6.16414645069413888699547509016, −4.30026099293465417809690979946, −3.90277145068512753118606146916, −2.53676864066042077082901779177, −1.40864519121509738458608423867,
0.57490128015347619171821733892, 2.50453516339212674287377002269, 4.41700022553385772334277498546, 5.42259412873009241545439419779, 6.35436406310115565615875567914, 7.37170606598318398420856886119, 7.64867495792063346925017572549, 8.765335982669256402004550709848, 9.220968817348667323199288076063, 10.38795564648361009113412217962