L(s) = 1 | + (0.707 − 0.707i)5-s + 3.46·7-s + (2.99 + 2.99i)11-s + (4.77 − 4.77i)13-s + 2.39i·17-s + (3.53 + 3.53i)19-s + 0.278i·23-s − 1.00i·25-s + (3.01 + 3.01i)29-s + 0.996i·31-s + (2.45 − 2.45i)35-s + (−7.22 − 7.22i)37-s − 2.39·41-s + (−6.32 + 6.32i)43-s + 4.37·47-s + ⋯ |
L(s) = 1 | + (0.316 − 0.316i)5-s + 1.31·7-s + (0.903 + 0.903i)11-s + (1.32 − 1.32i)13-s + 0.579i·17-s + (0.809 + 0.809i)19-s + 0.0581i·23-s − 0.200i·25-s + (0.559 + 0.559i)29-s + 0.178i·31-s + (0.414 − 0.414i)35-s + (−1.18 − 1.18i)37-s − 0.374·41-s + (−0.965 + 0.965i)43-s + 0.638·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.723950110\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.723950110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + (-2.99 - 2.99i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.77 + 4.77i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.39iT - 17T^{2} \) |
| 19 | \( 1 + (-3.53 - 3.53i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.278iT - 23T^{2} \) |
| 29 | \( 1 + (-3.01 - 3.01i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.996iT - 31T^{2} \) |
| 37 | \( 1 + (7.22 + 7.22i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + (6.32 - 6.32i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.37T + 47T^{2} \) |
| 53 | \( 1 + (7.49 - 7.49i)T - 53iT^{2} \) |
| 59 | \( 1 + (-7.26 - 7.26i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.978 - 0.978i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.87 + 4.87i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 - 2.98iT - 79T^{2} \) |
| 83 | \( 1 + (11.2 - 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.686131005675622810409138750859, −8.078304174164177268110821922758, −7.45039868880853602187753718233, −6.39712287856345702111691880489, −5.63407836187355078046119970325, −4.96346802587207090266165771535, −4.07532574826271189231019528550, −3.22010740281579174829970784390, −1.70301458161691013531830754632, −1.26207311497121534351141314938,
1.10340076920937700865457123329, 1.88448239442704855986501601605, 3.16006515984653145433908203103, 4.03199556087466592411652611889, 4.88746431909957963215102565492, 5.68883930767657267341978345074, 6.64175030399597745968874995881, 7.03337629676307607663769156741, 8.324977631460802456040253625076, 8.602465178817668613985528897014