| L(s) = 1 | − 18.3·5-s + 18i·13-s + 140. i·17-s + 213·25-s + 108.·29-s − 396i·37-s + 496. i·41-s + 343·49-s − 770.·53-s + 468i·61-s − 330. i·65-s − 592·73-s − 2.57e3i·85-s − 1.05e3i·89-s − 1.81e3·97-s + ⋯ |
| L(s) = 1 | − 1.64·5-s + 0.384i·13-s + 1.99i·17-s + 1.70·25-s + 0.697·29-s − 1.75i·37-s + 1.89i·41-s + 49-s − 1.99·53-s + 0.982i·61-s − 0.631i·65-s − 0.949·73-s − 3.28i·85-s − 1.25i·89-s − 1.90·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2008163356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2008163356\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 - 18iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 140. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 + 396iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 496. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 770.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 - 468iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 592T + 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
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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
−8.898406630325244494972926422988, −8.234499402695543138562486083800, −7.64417606209705253315451091447, −6.73422977769241169477222487143, −5.81770935679324149884641554258, −4.45731172314289042841743476570, −3.99793999974100759427451722994, −2.99981978123335405951308899064, −1.47445013071255324195551202670, −0.06517781382214689101433544798,
0.877705328346766892143239469982, 2.71305322515072008659668448732, 3.51010709223943012825312797950, 4.53555289415880912468798460296, 5.21054607338256129646492429297, 6.61893424048649174697038790167, 7.34590210258282990556875909939, 7.975702351633440608568939823126, 8.759828306022397451593700135276, 9.651695091214017559677917474645
