L(s) = 1 | + (1.16 + 1.16i)5-s − 3.74·7-s + (−1.64 + 1.64i)11-s + (0.645 + 0.645i)13-s + 6.57i·17-s + (−1.41 + 1.41i)19-s − 6i·23-s − 2.29i·25-s + (5.40 − 5.40i)29-s − 0.913i·31-s + (−4.35 − 4.35i)35-s + (−1 + i)37-s − 6.57·41-s + (−3.74 − 3.74i)43-s − 6·47-s + ⋯ |
L(s) = 1 | + (0.520 + 0.520i)5-s − 1.41·7-s + (−0.496 + 0.496i)11-s + (0.179 + 0.179i)13-s + 1.59i·17-s + (−0.324 + 0.324i)19-s − 1.25i·23-s − 0.458i·25-s + (1.00 − 1.00i)29-s − 0.164i·31-s + (−0.736 − 0.736i)35-s + (−0.164 + 0.164i)37-s − 1.02·41-s + (−0.570 − 0.570i)43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05166009186\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05166009186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.16 - 1.16i)T + 5iT^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 + (1.64 - 1.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.645 - 0.645i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.57iT - 17T^{2} \) |
| 19 | \( 1 + (1.41 - 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (-5.40 + 5.40i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.913iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 + (3.74 + 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (7.73 + 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.29 + 9.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (10.2 + 10.2i)T + 61iT^{2} \) |
| 67 | \( 1 + (10.8 - 10.8i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.70iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 + (10.9 + 10.9i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.412T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.568612921158056960957243786153, −8.103566103491969888583496572403, −6.69940631239401566334273387453, −6.54887125653599885576455703268, −5.80362515123925152300838504846, −4.60183063529493599136030759124, −3.67987286660209338930628076660, −2.78406933647825367410263572953, −1.88823607206170549291145534669, −0.01710634223782635842257189664,
1.35482153347143538614748886056, 2.91340981098919088820032023821, 3.27169502290219397328241772892, 4.70974171713930101221865481130, 5.40355120111846076599441815518, 6.18287046654060366943193207631, 6.95242181201583895912058016298, 7.70713220488160807439767228565, 8.876628242109851282264662144770, 9.215117444222944010415187453359