L(s) = 1 | + 3i·3-s + 10.4i·5-s − 6.42·7-s − 9·9-s − 61.6i·11-s − 64.8i·13-s − 31.2·15-s − 75.6·17-s − 10.3i·19-s − 19.2i·21-s + 156.·23-s + 16.3·25-s − 27i·27-s + 53.7i·29-s + 227.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.932i·5-s − 0.346·7-s − 0.333·9-s − 1.69i·11-s − 1.38i·13-s − 0.538·15-s − 1.07·17-s − 0.124i·19-s − 0.200i·21-s + 1.42·23-s + 0.131·25-s − 0.192i·27-s + 0.344i·29-s + 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.338461265\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.338461265\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 - 10.4iT - 125T^{2} \) |
| 7 | \( 1 + 6.42T + 343T^{2} \) |
| 11 | \( 1 + 61.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 64.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 75.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 227.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 70.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 89.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 388. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 324iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 920. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 995.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 791. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.87e3T + 9.12e5T^{2} \) |
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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
−10.94990260948332287058950926757, −10.08207012006162553646364478139, −8.930774948501923010197599156662, −8.178360241452430017176344057191, −6.85387930554797081692058598828, −6.03312912555073074319391660476, −4.94694672132284673435557998511, −3.36036134816789442073500953724, −2.88471657120935676578383145642, −0.49110437182719285386602376106,
1.27693681370273272238060629301, 2.42482035271289104433297996357, 4.30823829320853091540522653307, 4.96531360452056762308303483580, 6.56490473534773537384508999478, 7.05777120539566782326530413174, 8.342761985885729588086364141534, 9.184365952362803135646726775166, 9.872560341839893490451069111918, 11.26968821674925452091635688460