L(s) = 1 | + (−0.309 − 0.951i)2-s + (−1.61 − 1.17i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.618 + 1.90i)6-s + (−1.61 + 1.17i)7-s + (−2.42 − 1.76i)8-s + (0.309 + 0.951i)9-s − 0.999·10-s − 2·12-s + (−0.309 − 0.951i)13-s + (1.61 + 1.17i)14-s + (−1.61 + 1.17i)15-s + (−0.309 + 0.951i)16-s + (1.54 − 4.75i)17-s + (0.809 − 0.587i)18-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.934 − 0.678i)3-s + (0.404 − 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.252 + 0.776i)6-s + (−0.611 + 0.444i)7-s + (−0.858 − 0.623i)8-s + (0.103 + 0.317i)9-s − 0.316·10-s − 0.577·12-s + (−0.0857 − 0.263i)13-s + (0.432 + 0.314i)14-s + (−0.417 + 0.303i)15-s + (−0.0772 + 0.237i)16-s + (0.374 − 1.15i)17-s + (0.190 − 0.138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.242820 - 0.690982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.242820 - 0.690982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (1.61 - 1.17i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 4.75i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.85 - 3.52i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + (-7.28 + 5.29i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.42 + 1.76i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.04 + 2.93i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.78 - 8.55i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.47 - 4.70i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.85 - 5.70i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + (-3.70 + 11.4i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.61 - 1.17i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.09 - 9.51i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.85 - 5.70i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (4.01 + 12.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.50266869010809655950331514252, −12.08934263641003889101988723449, −11.24065252837820388532513249569, −10.01513309344965017522234829101, −9.166248815155562469040968109968, −7.35947481729673743681244868233, −6.24028125433049972033214549607, −5.36565454275731227588423987372, −2.95322586359254882503304593617, −0.995596748791839602684278442338,
3.19636470591864378394209702976, 4.99591922705429192385995306899, 6.27867351960954730284903549872, 7.01784743536703267942058993345, 8.414667415648191824996173831256, 9.859451708039210727580304439824, 10.74633008169944499573904372331, 11.61290363152341605320420917325, 12.69289239281387707343128558366, 14.09186230025149742741552000549