| L(s) = 1 | + 2.35·3-s + 2.56·5-s − 4.15i·7-s + 2.56·9-s − 2.33i·11-s + 4.29i·13-s + 6.04·15-s − 17-s + (3.68 + 2.33i)19-s − 9.79i·21-s + 1.82i·23-s + 1.56·25-s − 1.03·27-s + 1.20i·29-s + 1.32·31-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 1.14·5-s − 1.56i·7-s + 0.853·9-s − 0.703i·11-s + 1.19i·13-s + 1.55·15-s − 0.242·17-s + (0.844 + 0.535i)19-s − 2.13i·21-s + 0.379i·23-s + 0.312·25-s − 0.198·27-s + 0.223i·29-s + 0.237·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146784414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146784414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.68 - 2.33i)T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 4.15iT - 7T^{2} \) |
| 11 | \( 1 + 2.33iT - 11T^{2} \) |
| 13 | \( 1 - 4.29iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 - 1.82iT - 23T^{2} \) |
| 29 | \( 1 - 1.20iT - 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 + 5.49iT - 41T^{2} \) |
| 43 | \( 1 - 1.30iT - 43T^{2} \) |
| 47 | \( 1 + 6.99iT - 47T^{2} \) |
| 53 | \( 1 - 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 0.290T + 67T^{2} \) |
| 71 | \( 1 + 2.06T + 71T^{2} \) |
| 73 | \( 1 + 0.123T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.41iT - 97T^{2} \) |
| 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
−9.440279578336070835583130334183, −9.081738234203564075248299976513, −8.019188794819516002596355494790, −7.31758327359972982643036746338, −6.50837048545411335472581259291, −5.44218725535420681900103520224, −4.12929586865785206250528241022, −3.50190986633198472985929588709, −2.31010804065093342620906739905, −1.31255685127914307311767126837,
1.77757782025173532871210799920, 2.60327592727165964909293369584, 3.14054315114138498032335974684, 4.77867085699922091620480788561, 5.60014759285111154186900707785, 6.38262871508927122819468859088, 7.60741912083150485041624600494, 8.353119420695452026164942014469, 9.035948889058838956918688761930, 9.633735363911708196217766214153
