L(s) = 1 | − 2i·2-s + 3i·3-s − 4·4-s + (8.58 − 7.16i)5-s + 6·6-s + 7i·7-s + 8i·8-s − 9·9-s + (−14.3 − 17.1i)10-s + 30.6·11-s − 12i·12-s − 54.8i·13-s + 14·14-s + (21.4 + 25.7i)15-s + 16·16-s + 116. i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.767 − 0.640i)5-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (−0.453 − 0.542i)10-s + 0.840·11-s − 0.288i·12-s − 1.16i·13-s + 0.267·14-s + (0.370 + 0.443i)15-s + 0.250·16-s + 1.66i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.79570 - 0.840085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79570 - 0.840085i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (-8.58 + 7.16i)T \) |
| 7 | \( 1 - 7iT \) |
good | 11 | \( 1 - 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 116. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 195. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 47.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 83.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 280. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 538. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 555. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 312.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.06e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 279. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 643.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 578. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 648. iT - 9.12e5T^{2} \) |
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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
−11.94928036332232702494358319389, −10.48652370140049728255461176635, −10.08783734431758154857592263932, −8.897854734590767605525835531473, −8.310282702962232459473631959035, −6.26408209560378238230223438275, −5.29376823991481616178286800792, −4.13179153607396837746786202004, −2.69608329166519614258810054034, −1.06500562790745720517477535232,
1.32160384237932599394491304486, 3.12146957827705249853302875614, 4.82798584878994811954109720637, 6.12905566639893701644381612971, 6.93474756059210365853065720996, 7.64409001462231273293454499386, 9.300166182867803162648514154108, 9.651006477156212164706237461620, 11.28778346315728923826181285555, 11.96213460256418484216659271041