L(s) = 1 | − i·2-s + (0.581 + 1.63i)3-s − 4-s + 5-s + (1.63 − 0.581i)6-s + i·8-s + (−2.32 + 1.89i)9-s − i·10-s + 0.963i·11-s + (−0.581 − 1.63i)12-s − 1.98i·13-s + (0.581 + 1.63i)15-s + 16-s + 2.63·17-s + (1.89 + 2.32i)18-s + 5.16i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.335 + 0.941i)3-s − 0.5·4-s + 0.447·5-s + (0.666 − 0.237i)6-s + 0.353i·8-s + (−0.774 + 0.632i)9-s − 0.316i·10-s + 0.290i·11-s + (−0.167 − 0.470i)12-s − 0.551i·13-s + (0.150 + 0.421i)15-s + 0.250·16-s + 0.638·17-s + (0.447 + 0.547i)18-s + 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.703249004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.703249004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.581 - 1.63i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.963iT - 11T^{2} \) |
| 13 | \( 1 + 1.98iT - 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 5.16iT - 19T^{2} \) |
| 23 | \( 1 - 2.69iT - 23T^{2} \) |
| 29 | \( 1 - 9.90iT - 29T^{2} \) |
| 31 | \( 1 + 3.42iT - 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 13.1iT - 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 - 8.96iT - 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 + 0.408T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−9.702796930410045623984590495418, −9.169443673325905628829605602048, −8.219216141055500077482398645182, −7.52888871281109652990912276758, −6.01400648168680997374299704539, −5.36222422491104827736466311308, −4.44444515032285984500989258569, −3.50121388989806919308687339132, −2.74522124716738132414663932896, −1.47000870635958328568314464320,
0.68000636579766448864938358810, 2.10566037521461945258811447454, 3.15792910016573716123626035330, 4.43571564250811426746578750385, 5.47268853972977884893236327777, 6.37498564457060402758381453418, 6.78162429558372195539287698273, 7.83247690933422415545664351550, 8.318502414755594503557837191232, 9.281232379904535797891574060903