L(s) = 1 | + 3-s + 3.49i·5-s − i·7-s + 9-s + 0.708i·11-s + (−3.25 − 1.54i)13-s + 3.49i·15-s + 7.09·17-s + 0.311i·19-s − i·21-s + 7.88·23-s − 7.20·25-s + 27-s + 5.29·29-s − 7.29i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.56i·5-s − 0.377i·7-s + 0.333·9-s + 0.213i·11-s + (−0.903 − 0.429i)13-s + 0.901i·15-s + 1.72·17-s + 0.0714i·19-s − 0.218i·21-s + 1.64·23-s − 1.44·25-s + 0.192·27-s + 0.983·29-s − 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.530682428\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.530682428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (3.25 + 1.54i)T \) |
good | 5 | \( 1 - 3.49iT - 5T^{2} \) |
| 11 | \( 1 - 0.708iT - 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 - 0.311iT - 19T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 7.29iT - 31T^{2} \) |
| 37 | \( 1 - 1.41iT - 37T^{2} \) |
| 41 | \( 1 - 11.8iT - 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 6.11iT - 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 - 2.19iT - 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 - 9.17iT - 67T^{2} \) |
| 71 | \( 1 + 0.708iT - 71T^{2} \) |
| 73 | \( 1 - 5.21iT - 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 - 6.11iT - 83T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + 7.79iT - 97T^{2} \) |
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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
−8.232237393662451937922502534883, −7.68516586445053751793687702372, −7.13819484951644728198428945560, −6.55216100993964804158162316108, −5.60390919599833489985026058783, −4.70821571166999273360514374300, −3.67425556627742545450710058473, −2.96502372041420396358340647855, −2.52036269710935442140171226180, −1.08780364534416093695910680402,
0.78214073200931768790564703521, 1.62989610949423910201098777668, 2.78752788125647859082161319459, 3.59464917382501759920418033335, 4.74877944160219101693537120853, 5.03760276288810307368934303560, 5.84483975745379619917971981608, 6.96942581398352553197749776734, 7.65957885501340556251509262660, 8.358959511367050107845921446889