L(s) = 1 | + (4.80 + 1.39i)5-s − 12.3i·7-s − 4.62i·11-s + 1.26i·13-s − 22.9·17-s + 32.8·19-s − 0.797·23-s + (21.1 + 13.3i)25-s − 39.6i·29-s + 1.46·31-s + (17.2 − 59.4i)35-s − 44.9i·37-s + 39.1i·41-s − 36.3i·43-s − 61.0·47-s + ⋯ |
L(s) = 1 | + (0.960 + 0.278i)5-s − 1.76i·7-s − 0.420i·11-s + 0.0971i·13-s − 1.34·17-s + 1.73·19-s − 0.0346·23-s + (0.844 + 0.534i)25-s − 1.36i·29-s + 0.0471·31-s + (0.492 − 1.69i)35-s − 1.21i·37-s + 0.954i·41-s − 0.846i·43-s − 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.006480301\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006480301\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.80 - 1.39i)T \) |
good | 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 + 4.62iT - 121T^{2} \) |
| 13 | \( 1 - 1.26iT - 169T^{2} \) |
| 17 | \( 1 + 22.9T + 289T^{2} \) |
| 19 | \( 1 - 32.8T + 361T^{2} \) |
| 23 | \( 1 + 0.797T + 529T^{2} \) |
| 29 | \( 1 + 39.6iT - 841T^{2} \) |
| 31 | \( 1 - 1.46T + 961T^{2} \) |
| 37 | \( 1 + 44.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 13.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 58.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 29.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 130.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 48.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.3iT - 9.40e3T^{2} \) |
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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
−9.222605167940951776340095906432, −8.048807987642597258441800514968, −7.26378975189459004229191349672, −6.65266444964065162264266790044, −5.79811523799685423776345389190, −4.76474754237566476059043623383, −3.90048443358159489500199978507, −2.89925367126113545942272069168, −1.63706496457474233551731268821, −0.52601812243247307133689531896,
1.46852370178868437096667105704, 2.36658308425026452384267685482, 3.19305064832236312512418439041, 4.88645419496007410303136488557, 5.23578928437163503605060534177, 6.15965611604082566268720376448, 6.83143144824376652900046009477, 8.040176509013542717913043545844, 8.900639229693634596975768895384, 9.294238607658386172818244076924