L(s) = 1 | + 3i·3-s + 18.4i·5-s − 22.4·7-s − 9·9-s + 53.6i·11-s + 7.15i·13-s − 55.2·15-s + 39.6·17-s − 125. i·19-s − 67.2i·21-s − 99.1·23-s − 214.·25-s − 27i·27-s + 205. i·29-s + 147.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.64i·5-s − 1.21·7-s − 0.333·9-s + 1.47i·11-s + 0.152i·13-s − 0.951·15-s + 0.566·17-s − 1.51i·19-s − 0.698i·21-s − 0.898·23-s − 1.71·25-s − 0.192i·27-s + 1.31i·29-s + 0.854·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6906289647\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6906289647\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 - 18.4iT - 125T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 11 | \( 1 - 53.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 7.15iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 39.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 99.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 205. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 506.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.3iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 324iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 464. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 15.8iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 659.T + 9.12e5T^{2} \) |
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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
−11.30225976555928957787582592092, −10.32866005658714966905153584727, −9.938313674643854359198205903212, −9.043323646145954979444824081988, −7.36411157089547635280624042148, −6.89109352122068319829208800904, −5.90376624421957540451842809138, −4.41624287453501589321172149913, −3.27529135064497163698833997620, −2.44114199015463559474032336621,
0.24560500845527334316821758714, 1.28493397482986512940006365712, 3.08871217206903087785157406804, 4.28532495447804495880265413365, 5.91221435244200454627231044279, 6.01994780541879112933061756456, 7.87956553889069554842887383892, 8.329203946477962272240273477935, 9.387714408863935506144951377446, 10.10804277316003560236990867574