L(s) = 1 | + 1.41·2-s + 4.30·3-s + 2.00·4-s + 2.23i·5-s + 6.09·6-s − 1.47i·7-s + 2.82·8-s + 9.55·9-s + 3.16i·10-s − 6.04i·11-s + 8.61·12-s − 5.21·13-s − 2.08i·14-s + 9.63i·15-s + 4.00·16-s + 15.7i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.43·3-s + 0.500·4-s + 0.447i·5-s + 1.01·6-s − 0.210i·7-s + 0.353·8-s + 1.06·9-s + 0.316i·10-s − 0.549i·11-s + 0.717·12-s − 0.401·13-s − 0.149i·14-s + 0.642i·15-s + 0.250·16-s + 0.923i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.50319 + 0.197115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.50319 + 0.197115i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (2.58 + 22.8i)T \) |
good | 3 | \( 1 - 4.30T + 9T^{2} \) |
| 7 | \( 1 + 1.47iT - 49T^{2} \) |
| 11 | \( 1 + 6.04iT - 121T^{2} \) |
| 13 | \( 1 + 5.21T + 169T^{2} \) |
| 17 | \( 1 - 15.7iT - 289T^{2} \) |
| 19 | \( 1 - 4.82iT - 361T^{2} \) |
| 29 | \( 1 + 23.4T + 841T^{2} \) |
| 31 | \( 1 + 20.4T + 961T^{2} \) |
| 37 | \( 1 - 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 38.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 32.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154. iT - 9.40e3T^{2} \) |
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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
−12.26416039428599729892952213699, −10.99500616372114808632480950924, −10.11001031318905869074194469366, −8.911726367472106945876055037027, −8.005745351160533964688404349985, −7.06178841392708776897231137193, −5.80591458399688865510423525226, −4.15757545772656088643431420066, −3.25048400981148179439600801926, −2.10531419447362216751149016206,
1.98484653869137568546405023694, 3.13983258056870335695236089862, 4.34194170568847176470301718332, 5.52483753706758343949500359983, 7.16783440963346792715455669187, 7.88250427250018570859625836376, 9.138347776505082470232370047812, 9.670712369447652528099194898927, 11.18504609871986250428365107913, 12.25928599556959807932678697201