| L(s) = 1 | + 2·7-s − 4i·11-s − 6·17-s + 4i·19-s + 4·23-s − 6i·29-s − 10·31-s + 4i·37-s − 10·41-s + 4i·43-s + 4·47-s − 3·49-s + 10i·53-s − 8i·59-s + 8i·61-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20i·11-s − 1.45·17-s + 0.917i·19-s + 0.834·23-s − 1.11i·29-s − 1.79·31-s + 0.657i·37-s − 1.56·41-s + 0.609i·43-s + 0.583·47-s − 0.428·49-s + 1.37i·53-s − 1.04i·59-s + 1.02i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5362681447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5362681447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.254257042270967831663337016219, −7.56111487484050192244346509082, −6.77895682556346781878408506442, −6.04155913981755672752569515406, −5.42069744656772704215736941359, −4.62144264495971830175734861054, −3.89166533939945776749928390210, −3.05995143940834878176874049577, −2.09748199427058126996183665958, −1.20077452636665834654773494768,
0.12422844191681540470734810695, 1.66530457129495283113380759426, 2.15036639743811385067344143546, 3.26056773834647658783774046257, 4.18982472380247506598357557140, 4.94224206191921137572447356936, 5.23221646986184009280502399006, 6.45888692714880299680654933872, 7.11936729598252928457586900513, 7.40348470940267704055921708505
