L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.0135 + 0.0941i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (0.0552 − 0.0775i)6-s + (0.370 − 1.52i)7-s + (0.654 − 0.755i)8-s + (0.950 − 0.279i)9-s + (−0.928 + 0.371i)10-s + (−0.0934 + 0.0180i)12-s + (−1.32 + 0.849i)14-s + (0.0913 + 0.0268i)15-s + (−0.995 + 0.0950i)16-s + (−0.880 − 0.454i)18-s + (0.928 + 0.371i)20-s + (0.148 + 0.0142i)21-s + ⋯ |
L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.0135 + 0.0941i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (0.0552 − 0.0775i)6-s + (0.370 − 1.52i)7-s + (0.654 − 0.755i)8-s + (0.950 − 0.279i)9-s + (−0.928 + 0.371i)10-s + (−0.0934 + 0.0180i)12-s + (−1.32 + 0.849i)14-s + (0.0913 + 0.0268i)15-s + (−0.995 + 0.0950i)16-s + (−0.880 − 0.454i)18-s + (0.928 + 0.371i)20-s + (0.148 + 0.0142i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8791990666\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8791990666\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.723 + 0.690i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.888 - 0.458i)T \) |
good | 3 | \( 1 + (-0.0135 - 0.0941i)T + (-0.959 + 0.281i)T^{2} \) |
| 7 | \( 1 + (-0.370 + 1.52i)T + (-0.888 - 0.458i)T^{2} \) |
| 11 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 13 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 17 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 19 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 23 | \( 1 + (0.786 - 0.618i)T + (0.235 - 0.971i)T^{2} \) |
| 29 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.581 + 0.299i)T + (0.580 - 0.814i)T^{2} \) |
| 43 | \( 1 + (-1.49 - 0.961i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (1.82 + 0.729i)T + (0.723 + 0.690i)T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.481 + 0.676i)T + (-0.327 - 0.945i)T^{2} \) |
| 71 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 73 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 79 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 83 | \( 1 + (-0.469 + 0.0448i)T + (0.981 - 0.189i)T^{2} \) |
| 89 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.660136518475923240991894129209, −9.005543082387840345005325208178, −7.955321394059655124574688543696, −7.43742701290902232950158578747, −6.55711403658371262533670600671, −5.07695533826108704195732736919, −4.20319104173425804016971890709, −3.58812742695941104656405732769, −1.83379171892883669375844378271, −1.03817026868125200290738841747,
1.86712990653444443415594071314, 2.50562028299660258846110368424, 4.25773283245910289429945559910, 5.43112028741050541930816110152, 6.03311619915857372009607309740, 6.78224524334774114674757044530, 7.72719311009840126150610350695, 8.280581382049496345084024899815, 9.405435512627883931410973358026, 9.730584345395066734102995850609