L(s) = 1 | + i·2-s + (−0.618 + 1.61i)3-s − 4-s + (−1.61 − 0.618i)6-s + (−0.381 + 2.61i)7-s − i·8-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (0.618 − 1.61i)12-s + 3.23i·13-s + (−2.61 − 0.381i)14-s + 16-s − 0.763·17-s + (2.00 − 2.23i)18-s + 0.472i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.356 + 0.934i)3-s − 0.5·4-s + (−0.660 − 0.252i)6-s + (−0.144 + 0.989i)7-s − 0.353i·8-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (0.178 − 0.467i)12-s + 0.897i·13-s + (−0.699 − 0.102i)14-s + 0.250·16-s − 0.185·17-s + (0.471 − 0.527i)18-s + 0.108i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7328633073\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7328633073\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.381 - 2.61i)T \) |
good | 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 0.472iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 5.70iT - 29T^{2} \) |
| 31 | \( 1 - 7.23iT - 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2.76iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 2.76iT - 71T^{2} \) |
| 73 | \( 1 + 6.76iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.23iT - 97T^{2} \) |
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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
−10.14870212424882889376197975683, −9.593274806164379549035724775503, −8.897128894602978971930438835404, −8.145196080897553356214580767023, −6.80948714588872995750633334259, −6.33133131518256237060506486834, −5.13928255959841595481325715070, −4.69416515476614367899637449225, −3.60181022102132425019819931696, −2.16756668792110497854861664515,
0.35789036118455111709010326276, 1.40699466418107158754766916260, 2.89100182024569108966458522074, 3.70614113597681864478282335179, 5.09411750310184589743828675343, 5.89935941887595890932446585988, 6.86115400907298562547086373970, 7.80061495498175340239336679850, 8.393593610158102822196489113398, 9.455985958683693746162590004894