L(s) = 1 | − i·2-s − 2.44i·3-s − 4-s + (−0.224 + 2.22i)5-s − 2.44·6-s − i·7-s + i·8-s − 2.99·9-s + (2.22 + 0.224i)10-s + 4.89·11-s + 2.44i·12-s + 0.449i·13-s − 14-s + (5.44 + 0.550i)15-s + 16-s + 2i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.41i·3-s − 0.5·4-s + (−0.100 + 0.994i)5-s − 0.999·6-s − 0.377i·7-s + 0.353i·8-s − 0.999·9-s + (0.703 + 0.0710i)10-s + 1.47·11-s + 0.707i·12-s + 0.124i·13-s − 0.267·14-s + (1.40 + 0.142i)15-s + 0.250·16-s + 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.594372 - 0.657441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.594372 - 0.657441i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.224 - 2.22i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 - 6.89iT - 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 + 0.898iT - 47T^{2} \) |
| 53 | \( 1 - 1.10iT - 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.89iT - 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
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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
−14.08526841533397932505097033534, −13.30914428357588708925777978210, −12.12441150829848094183317813801, −11.38654572036831701617938777479, −10.14426114234254611513281680593, −8.555439577994667383484011730629, −7.16680802511913941244908265704, −6.31546491003053097024332362523, −3.74623558901870409569966711662, −1.82896760108220512789219906716,
4.04853735820217459631622764883, 4.92316433075312663206256433053, 6.41637525834994849944410655192, 8.513163680692904578369236277240, 9.099332830909959217552901262909, 10.20015222730524925012267609842, 11.70172461178631872541256264227, 12.88787510282672316243007210570, 14.42659898044210509523723155384, 15.08517839053954537072533383340