L(s) = 1 | − i·2-s + (1.13 + 0.658i)3-s − 4-s + (−1.72 − 1.42i)5-s + (0.658 − 1.13i)6-s + (−3.46 + 2.00i)7-s + i·8-s + (−0.633 − 1.09i)9-s + (−1.42 + 1.72i)10-s − 4.54·11-s + (−1.13 − 0.658i)12-s + (0.344 − 0.198i)13-s + (2.00 + 3.46i)14-s + (−1.02 − 2.75i)15-s + 16-s + (−5.13 + 2.96i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.658 + 0.379i)3-s − 0.5·4-s + (−0.771 − 0.636i)5-s + (0.268 − 0.465i)6-s + (−1.30 + 0.756i)7-s + 0.353i·8-s + (−0.211 − 0.365i)9-s + (−0.450 + 0.545i)10-s − 1.36·11-s + (−0.329 − 0.189i)12-s + (0.0954 − 0.0550i)13-s + (0.534 + 0.926i)14-s + (−0.265 − 0.711i)15-s + 0.250·16-s + (−1.24 + 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0372871 + 0.185644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0372871 + 0.185644i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (1.72 + 1.42i)T \) |
| 43 | \( 1 + (-0.0743 + 6.55i)T \) |
good | 3 | \( 1 + (-1.13 - 0.658i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (3.46 - 2.00i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 + (-0.344 + 0.198i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (5.13 - 2.96i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.29 + 5.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.205 + 0.118i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.62 - 6.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.00 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.13 + 2.96i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.91T + 41T^{2} \) |
| 47 | \( 1 - 7.24iT - 47T^{2} \) |
| 53 | \( 1 + (-3.99 - 2.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.57 - 2.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.50 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.61 - 4.97i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.74 + 9.94i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.52 + 0.881i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.40 - 12.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.92iT - 97T^{2} \) |
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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
−10.60550302785861968065162655317, −9.641387464955946340355079745823, −8.842622136634519983626595605757, −8.444429227203045356640660096398, −7.01492631679978767603952314063, −5.63645233593518382849938534602, −4.50213326885105708653859597978, −3.32679609699092085053788199520, −2.62463730829625428393263324685, −0.10352631898681243841110563732,
2.73698584694218535339866465185, 3.59708227941249503273224801157, 4.96759421645839883818236669320, 6.40028525637771419459848523119, 7.14621160595494964335060553075, 7.86788606208693208019356552959, 8.579619009731374777029165714283, 9.958417217202214540556941026651, 10.47019389770147007386301733766, 11.71489953659753999118676087717