L(s) = 1 | − i·2-s − 0.646i·3-s − 4-s − 3.09i·5-s − 0.646·6-s + (−2.44 + i)7-s + i·8-s + 2.58·9-s − 3.09·10-s + (−1.79 − 2.79i)11-s + 0.646i·12-s − 0.646·13-s + (1 + 2.44i)14-s − 2·15-s + 16-s + 3.74·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.373i·3-s − 0.5·4-s − 1.38i·5-s − 0.263·6-s + (−0.925 + 0.377i)7-s + 0.353i·8-s + 0.860·9-s − 0.978·10-s + (−0.540 − 0.841i)11-s + 0.186i·12-s − 0.179·13-s + (0.267 + 0.654i)14-s − 0.516·15-s + 0.250·16-s + 0.907·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.458677 - 0.883036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.458677 - 0.883036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 + (2.44 - i)T \) |
| 11 | \( 1 + (1.79 + 2.79i)T \) |
good | 3 | \( 1 + 0.646iT - 3T^{2} \) |
| 5 | \( 1 + 3.09iT - 5T^{2} \) |
| 13 | \( 1 + 0.646T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 1.58iT - 29T^{2} \) |
| 31 | \( 1 - 8.64iT - 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 + 7.16iT - 43T^{2} \) |
| 47 | \( 1 - 9.93iT - 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 0.646iT - 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−12.76260955664371004669504647534, −11.96283610325945082756514179657, −10.55466554067785530502279507155, −9.498887146374698098197804506881, −8.744165601952647794709150904047, −7.51991935641800979363640080866, −5.85554968707577883428354475765, −4.72637924346504236403334251266, −3.13191279228066023210381083585, −1.10198934929169141204728050193,
3.03321474669947362489101445092, 4.36189856840803510619947827250, 5.97273854245482618408482923496, 7.11297307059603508200320801446, 7.59094116905407526497685219598, 9.644581831657590298578839782687, 9.965757449087561524478524734369, 11.03833595416215798126071758866, 12.59433789464731410745975141423, 13.39505171466014102806473411538