L(s) = 1 | − 2-s + 0.396·3-s − 3·4-s + 2.23i·5-s − 0.396·6-s − 8.16i·7-s + 7·8-s − 8.84·9-s − 2.23i·10-s − 18.9i·11-s − 1.18·12-s − 18.6·13-s + 8.16i·14-s + 0.885i·15-s + 5·16-s + 5.57i·17-s + ⋯ |
L(s) = 1 | − 0.5·2-s + 0.132·3-s − 0.750·4-s + 0.447i·5-s − 0.0660·6-s − 1.16i·7-s + 0.875·8-s − 0.982·9-s − 0.223i·10-s − 1.72i·11-s − 0.0990·12-s − 1.43·13-s + 0.583i·14-s + 0.0590i·15-s + 0.312·16-s + 0.327i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.212670 - 0.423538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.212670 - 0.423538i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (18.4 + 13.7i)T \) |
good | 2 | \( 1 + T + 4T^{2} \) |
| 3 | \( 1 - 0.396T + 9T^{2} \) |
| 7 | \( 1 + 8.16iT - 49T^{2} \) |
| 11 | \( 1 + 18.9iT - 121T^{2} \) |
| 13 | \( 1 + 18.6T + 169T^{2} \) |
| 17 | \( 1 - 5.57iT - 289T^{2} \) |
| 19 | \( 1 + 5.25iT - 361T^{2} \) |
| 29 | \( 1 - 11.6T + 841T^{2} \) |
| 31 | \( 1 - 49.5T + 961T^{2} \) |
| 37 | \( 1 - 29.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 57.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 15.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 27.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 35.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 0.377T + 5.32e3T^{2} \) |
| 79 | \( 1 + 102. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 49.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 69.6iT - 9.40e3T^{2} \) |
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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
−13.30237057111521590518999548538, −11.71939088047803896402452945730, −10.63902523332006551031143120020, −9.856784683828807411552348718661, −8.521451356642158963922695634164, −7.80355792889313172818936540948, −6.28288298550766513242570000846, −4.68126401215102877797306390303, −3.16588245076992237593994637047, −0.38970212509846125275750858787,
2.31405819151971534130614912126, 4.56466438561201244349145874715, 5.52672603610774314416552750227, 7.44091616050910172740891091584, 8.498562252877243683532421923809, 9.413569887747486527880429154900, 10.07821953761592870143475109749, 12.01522109523490172893201210524, 12.34164426235018446153084098861, 13.75677672339866887875099155256