L(s) = 1 | + 0.951i·2-s + 5.19i·3-s + 15.0·4-s − 27.2i·5-s − 4.94·6-s + 0.640i·7-s + 29.5i·8-s − 27·9-s + 25.9·10-s − 120. i·11-s + 78.4i·12-s − 247. i·13-s − 0.609·14-s + 141.·15-s + 213.·16-s − 172.·17-s + ⋯ |
L(s) = 1 | + 0.237i·2-s + 0.577i·3-s + 0.943·4-s − 1.09i·5-s − 0.137·6-s + 0.0130i·7-s + 0.462i·8-s − 0.333·9-s + 0.259·10-s − 0.996i·11-s + 0.544i·12-s − 1.46i·13-s − 0.00310·14-s + 0.629·15-s + 0.833·16-s − 0.596·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.957710333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.957710333\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19iT \) |
| 67 | \( 1 + (-3.94e3 + 2.13e3i)T \) |
good | 2 | \( 1 - 0.951iT - 16T^{2} \) |
| 5 | \( 1 + 27.2iT - 625T^{2} \) |
| 7 | \( 1 - 0.640iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 120. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 247. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 172.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 265.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 339.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 988.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.63e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.47e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.32e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.16e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.86e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 5.11e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 938.T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.00e3iT - 1.38e7T^{2} \) |
| 71 | \( 1 - 4.25e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.52e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.01e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.12e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 829.T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.06e4iT - 8.85e7T^{2} \) |
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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
−11.41622573370375038353205709626, −10.79739464247420036674053875065, −9.619021112702079318002947435922, −8.460746243879391194983357208491, −7.78758684715350388896629264260, −6.10702425477143560992343704093, −5.44277071152435075256035923059, −3.99071578141157207625567444075, −2.50213479970678519999327223027, −0.64973397331206724765504787517,
1.77572790098419929650234762857, 2.59387076399362640725992373452, 4.14027821456941007369667464446, 6.10718793427170577283436493293, 6.92670359450541524401041144813, 7.42534021881560210387027827249, 9.027317596445813494532626916457, 10.29228521841996591576710533044, 11.04765679531444091391176613089, 11.87064504906749848808241535586