L(s) = 1 | + (−2.05 + 1.18i)5-s + (4.05 − 7.02i)7-s + (−17.6 − 10.1i)11-s + (3.05 + 5.29i)13-s − 17.9i·17-s − 9.11·19-s + (−29.0 + 16.7i)23-s + (−9.67 + 16.7i)25-s + (14.4 + 8.31i)29-s + (11.1 + 19.3i)31-s + 19.2i·35-s + 50.4·37-s + (−29.9 + 17.3i)41-s + (11.5 − 19.9i)43-s + (33.1 + 19.1i)47-s + ⋯ |
L(s) = 1 | + (−0.411 + 0.237i)5-s + (0.579 − 1.00i)7-s + (−1.60 − 0.924i)11-s + (0.235 + 0.407i)13-s − 1.05i·17-s − 0.479·19-s + (−1.26 + 0.729i)23-s + (−0.387 + 0.670i)25-s + (0.496 + 0.286i)29-s + (0.360 + 0.624i)31-s + 0.551i·35-s + 1.36·37-s + (−0.730 + 0.421i)41-s + (0.267 − 0.463i)43-s + (0.705 + 0.407i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7035483925\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7035483925\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.05 - 1.18i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.05 + 7.02i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (17.6 + 10.1i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.05 - 5.29i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 17.9iT - 289T^{2} \) |
| 19 | \( 1 + 9.11T + 361T^{2} \) |
| 23 | \( 1 + (29.0 - 16.7i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.4 - 8.31i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.9 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11.5 + 19.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-33.1 - 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (2.96 - 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.1 - 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (3.14 + 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-42.2 + 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (33.1 + 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.3 - 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.371053866061854889041237185883, −8.306945726575049008186675057425, −7.77388664668028908862068282295, −7.20456457604045536364605883655, −6.11213311510625734424354166258, −5.21208870703570201545509928836, −4.36501552705546187632614420598, −3.45060601415469129935581441389, −2.46108088740771787740585238906, −1.01125906820547450643800429045,
0.20760887160761254164935793345, 2.00147730258222819063180793169, 2.59204200890360450930315458894, 4.07121394086616238684426293341, 4.76857649865393333233312737094, 5.67215948941419450408283141530, 6.34189395484854486795895805058, 7.71240530600281150655740822215, 8.120699183275643085851911555361, 8.625914908707607682457055299819