| L(s) = 1 | + 2.44i·5-s + 3.46i·7-s − 2.82·11-s + 3.46·13-s − 1.41i·17-s + 4i·19-s + 4.89·23-s − 0.999·25-s + 2.44i·29-s + 3.46i·31-s − 8.48·35-s + 1.41i·41-s + 8i·43-s + 4.89·47-s − 4.99·49-s + ⋯ |
| L(s) = 1 | + 1.09i·5-s + 1.30i·7-s − 0.852·11-s + 0.960·13-s − 0.342i·17-s + 0.917i·19-s + 1.02·23-s − 0.199·25-s + 0.454i·29-s + 0.622i·31-s − 1.43·35-s + 0.220i·41-s + 1.21i·43-s + 0.714·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.415178765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415178765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.200806064306359693762186658568, −8.591976649893225392959008431495, −7.78571675343041723351086418188, −6.96411350131268180210297403968, −6.11518979184510579579460231321, −5.58093968608192679345298311555, −4.59749778503065992265910527062, −3.15891975779782326235472552200, −2.88777570172175907567968196737, −1.63504159183850030759332373133,
0.50458069272651905907396621299, 1.40663900171348385849899194011, 2.87659679986858846645610475653, 3.99554114788209599185310858396, 4.61154467708022864349819379714, 5.42128221657938213806829847281, 6.35411451809371509146001880576, 7.33541784921936511322611394580, 7.86110018961938054852364133128, 8.825990884973199206743895679054
