L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.51 − 0.838i)3-s + 1.00i·4-s + (−2.23 + 0.0661i)5-s + (0.479 + 1.66i)6-s + (2.39 − 2.39i)7-s + (0.707 − 0.707i)8-s + (1.59 + 2.54i)9-s + (1.62 + 1.53i)10-s − 3.77i·11-s + (0.838 − 1.51i)12-s + (3.87 + 3.87i)13-s − 3.38·14-s + (3.44 + 1.77i)15-s − 1.00·16-s + (2.58 + 2.58i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.875 − 0.483i)3-s + 0.500i·4-s + (−0.999 + 0.0295i)5-s + (0.195 + 0.679i)6-s + (0.905 − 0.905i)7-s + (0.250 − 0.250i)8-s + (0.531 + 0.846i)9-s + (0.514 + 0.484i)10-s − 1.13i·11-s + (0.241 − 0.437i)12-s + (1.07 + 1.07i)13-s − 0.905·14-s + (0.889 + 0.457i)15-s − 0.250·16-s + (0.626 + 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.170662 - 0.614568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.170662 - 0.614568i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.51 + 0.838i)T \) |
| 5 | \( 1 + (2.23 - 0.0661i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-2.39 + 2.39i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.77iT - 11T^{2} \) |
| 13 | \( 1 + (-3.87 - 3.87i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.58 - 2.58i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.80iT - 19T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 7.09T + 31T^{2} \) |
| 37 | \( 1 + (-4.68 + 4.68i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.03iT - 41T^{2} \) |
| 43 | \( 1 + (2.86 + 2.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.40 + 3.40i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.15 + 1.15i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.993T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + (-1.09 + 1.09i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.26iT - 71T^{2} \) |
| 73 | \( 1 + (-2.14 - 2.14i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.59iT - 79T^{2} \) |
| 83 | \( 1 + (-5.20 + 5.20i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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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
−10.69063225138074392317484668748, −9.134060682656348703884463020636, −8.314534679283945611574824989212, −7.50797005784102107449711542407, −6.86315645916982121176822091604, −5.60651917866857378996855655252, −4.35296163082227586553226640566, −3.61221975522759745268142617310, −1.66373573396706601116580450479, −0.50742543765063571413499760621,
1.41993024504378631451605115336, 3.51096496112705305026026392946, 4.68837436286328034854385112996, 5.45095150538377293495805331594, 6.27335076673915599515908167430, 7.63454774316645863179178345550, 7.991137933356820672974750746122, 9.104533162383959708268624850596, 9.969114695865977323527699484267, 10.85898023454239816313518242769