L(s) = 1 | − 5.24·3-s − 2.23i·5-s + 3.70i·7-s + 18.4·9-s − 3.04i·11-s + 17.9·13-s + 11.7i·15-s − 20.3i·17-s + 6.01i·19-s − 19.4i·21-s + (14.1 − 18.1i)23-s − 5.00·25-s − 49.5·27-s + 0.460·29-s + 1.09·31-s + ⋯ |
L(s) = 1 | − 1.74·3-s − 0.447i·5-s + 0.529i·7-s + 2.05·9-s − 0.277i·11-s + 1.38·13-s + 0.781i·15-s − 1.19i·17-s + 0.316i·19-s − 0.924i·21-s + (0.616 − 0.787i)23-s − 0.200·25-s − 1.83·27-s + 0.0158·29-s + 0.0351·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.052038438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.052038438\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-14.1 + 18.1i)T \) |
good | 3 | \( 1 + 5.24T + 9T^{2} \) |
| 7 | \( 1 - 3.70iT - 49T^{2} \) |
| 11 | \( 1 + 3.04iT - 121T^{2} \) |
| 13 | \( 1 - 17.9T + 169T^{2} \) |
| 17 | \( 1 + 20.3iT - 289T^{2} \) |
| 19 | \( 1 - 6.01iT - 361T^{2} \) |
| 29 | \( 1 - 0.460T + 841T^{2} \) |
| 31 | \( 1 - 1.09T + 961T^{2} \) |
| 37 | \( 1 - 31.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 74.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 12.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 53.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 90.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 57.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 63.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 10.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 46.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 62.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 91.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 156. iT - 9.40e3T^{2} \) |
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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
−9.053469572150520870103087351307, −8.232528353004262249559937793344, −7.17020013273120856358634921169, −6.27821234238304551712454669931, −5.87853397203295636888168935016, −4.98741600996749171759921624966, −4.40656403860540942479178376531, −3.07462878266693863875070033953, −1.44033872259926189606366123450, −0.57044558046687152031741738236,
0.75202394198318303980106002389, 1.75972660296063292128496860676, 3.57631778659562379094417657192, 4.22628428740989512645327408513, 5.34054916346449349970753750260, 5.90989149006257794300270108010, 6.70114097451033182873024889209, 7.20757980588453680269558255139, 8.269289540437356458949026751532, 9.323112660446871589852277749432