L(s) = 1 | − 2·13-s + (4 + 6.92i)19-s + (2.5 − 4.33i)25-s + (−2 + 3.46i)31-s + (5 + 8.66i)37-s + 8·43-s + (7 + 12.1i)61-s + (8 − 13.8i)67-s + (−5 + 8.66i)73-s + (2 + 3.46i)79-s − 14·97-s + (10 + 17.3i)103-s + (−1 + 1.73i)109-s + ⋯ |
L(s) = 1 | − 0.554·13-s + (0.917 + 1.58i)19-s + (0.5 − 0.866i)25-s + (−0.359 + 0.622i)31-s + (0.821 + 1.42i)37-s + 1.21·43-s + (0.896 + 1.55i)61-s + (0.977 − 1.69i)67-s + (−0.585 + 1.01i)73-s + (0.225 + 0.389i)79-s − 1.42·97-s + (0.985 + 1.70i)103-s + (−0.0957 + 0.165i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.589925061\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.589925061\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.530044344664672823687370209582, −8.516798118005930908986577574619, −7.86147694135727985556909256989, −7.07050729325715355674057844797, −6.15921684507740690464974393958, −5.36215989589383442813981310633, −4.45113063728847987930472033292, −3.46801658604687274419207993817, −2.45455288832451320950951149337, −1.14673234245407380756870781757,
0.69335406972705442880699528951, 2.22379300296538333079826554310, 3.14774433695310256136608703272, 4.26571316277168790413314224553, 5.13500414511092058323493125901, 5.88020959733861392847866171818, 7.07345698940820125811355153222, 7.39745551303209634332365244822, 8.472291674775557307128830219238, 9.361189171956335614796828198990