L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − i·12-s + i·13-s + 16-s − 6i·17-s + i·18-s + 4i·23-s − 24-s + 26-s − i·27-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.277i·13-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 0.834i·23-s − 0.204·24-s + 0.196·26-s − 0.192i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.577941544\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.577941544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.324829414365897403534583934842, −8.616922852978376070472245639819, −7.67974513181660500879309604141, −6.79079837994010600856173324279, −5.68444626772792164123977746234, −4.89853320263521452178116321586, −4.15707191570699309875231884706, −3.16236242952289672021194959278, −2.34576781878962605888678853675, −0.837038914943169770057565526587,
0.872231426305974713810506079661, 2.24604823741025748567118278636, 3.45813428929516679023275408846, 4.47907657810209676321139527920, 5.38195764772268698999721999460, 6.33601106139981271770406232860, 6.68956492523292333560510877869, 7.76155103767951533411174428588, 8.345238335244018553032912949153, 8.873004737428168776880637713402