L(s) = 1 | + 2-s + 4-s + (−0.707 − 0.707i)5-s + i·7-s + 8-s + i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s + i·18-s + (1.41 + 1.41i)19-s + (−0.707 − 0.707i)20-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)26-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + (−0.707 − 0.707i)5-s + i·7-s + 8-s + i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s + i·18-s + (1.41 + 1.41i)19-s + (−0.707 − 0.707i)20-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.135885683\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.135885683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1 + i)T - iT^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.585154755341563990106239624428, −7.924077542552287779973685778957, −7.45381032857878614910756251525, −6.41395999092546758011073058369, −5.54836700878340647908528228622, −5.05961432111678777325759964453, −4.34485062934813516604677373640, −3.49313771771896681891951132144, −2.43609213270872656410953348102, −1.66232066301673123775747521073,
0.910706297014820153260356641748, 2.52943712198888682254080657087, 3.37200176047433054491382579043, 3.80926429032578453526854788226, 4.72219311410426133692717007630, 5.54395367591771906483163770696, 6.48136347046059531427752747270, 7.20733534094975036539282178375, 7.42810941086616045529009085760, 8.320218487681190207632552834164