L(s) = 1 | + 2.55i·3-s − 9.86·5-s − 2.64i·7-s + 2.47·9-s − 13.1i·11-s + 5.86·13-s − 25.2i·15-s − 0.570·17-s + 15.6i·19-s + 6.75·21-s + 16.4i·23-s + 72.3·25-s + 29.3i·27-s + 29.7·29-s − 54.8i·31-s + ⋯ |
L(s) = 1 | + 0.851i·3-s − 1.97·5-s − 0.377i·7-s + 0.274·9-s − 1.19i·11-s + 0.451·13-s − 1.68i·15-s − 0.0335·17-s + 0.824i·19-s + 0.321·21-s + 0.716i·23-s + 2.89·25-s + 1.08i·27-s + 1.02·29-s − 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.139193989\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139193989\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 - 2.55iT - 9T^{2} \) |
| 5 | \( 1 + 9.86T + 25T^{2} \) |
| 11 | \( 1 + 13.1iT - 121T^{2} \) |
| 13 | \( 1 - 5.86T + 169T^{2} \) |
| 17 | \( 1 + 0.570T + 289T^{2} \) |
| 19 | \( 1 - 15.6iT - 361T^{2} \) |
| 23 | \( 1 - 16.4iT - 529T^{2} \) |
| 29 | \( 1 - 29.7T + 841T^{2} \) |
| 31 | \( 1 + 54.8iT - 961T^{2} \) |
| 37 | \( 1 - 42.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 0.773T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.65T + 2.80e3T^{2} \) |
| 59 | \( 1 + 42.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 92.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 9.97T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 5.79T + 7.92e3T^{2} \) |
| 97 | \( 1 - 103.T + 9.40e3T^{2} \) |
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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
−11.07637318309122255649351053384, −10.14569048514956629265370948787, −8.997559026216254347259616625256, −8.072559381031473064986624822255, −7.50389464511143415499158048131, −6.17927133782148962259596848137, −4.73404147853382012332544751668, −3.89419746069204789473863931089, −3.35218814684648177528320675853, −0.68621327315187088471860353336,
0.954609677208884970357914994849, 2.69133443278862507614580771475, 4.08652792833662535520904490072, 4.85513117248895812059897300393, 6.74553986748756376865016633970, 7.09521633860468985050262159843, 8.095110550251668576051706185531, 8.688541988974251086566467820873, 10.09725157011873798189209886507, 11.16559489111803677349714954597