L(s) = 1 | + 3i·2-s − 3i·3-s − 4-s + 9·6-s − 7i·7-s + 21i·8-s − 9·9-s − 36·11-s + 3i·12-s − 34i·13-s + 21·14-s − 71·16-s − 42i·17-s − 27i·18-s + 124·19-s + ⋯ |
L(s) = 1 | + 1.06i·2-s − 0.577i·3-s − 0.125·4-s + 0.612·6-s − 0.377i·7-s + 0.928i·8-s − 0.333·9-s − 0.986·11-s + 0.0721i·12-s − 0.725i·13-s + 0.400·14-s − 1.10·16-s − 0.599i·17-s − 0.353i·18-s + 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.199350181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199350181\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 2 | \( 1 - 3iT - 8T^{2} \) |
| 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 42iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 124T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160T + 2.97e4T^{2} \) |
| 37 | \( 1 + 398iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 318T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 240iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 498iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 132T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398T + 2.26e5T^{2} \) |
| 67 | \( 1 + 92iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 720T + 3.57e5T^{2} \) |
| 73 | \( 1 + 502iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 204iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 354T + 7.04e5T^{2} \) |
| 97 | \( 1 - 286iT - 9.12e5T^{2} \) |
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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
−10.32257245525055849242410444051, −9.143439904311811354957570134750, −8.079302956043925042421468521326, −7.45558017372683787280519828667, −6.91869649410433302313298807450, −5.56199206591429851803549688016, −5.23618932718648716172114894495, −3.37123498533767392642043421118, −2.11091259680074434324615154753, −0.34937687866765288249035281407,
1.45223355996482604762642215167, 2.69734339914058008430854870133, 3.55039658846958212328398534473, 4.73330100481057940440338863548, 5.78029321645497964324701007115, 6.99400254499154043103699204717, 8.090003120939171846755625404253, 9.251331029544100009051416102202, 9.865879719944274928991822775243, 10.66193279970819223325488061387