| L(s) = 1 | + 0.732i·2-s + 1.46·4-s − 4.73i·7-s + 2.53i·8-s − 5.73·11-s − 1.46i·13-s + 3.46·14-s + 1.07·16-s − 2.73i·17-s − 4.46·19-s − 4.19i·22-s + 3.46i·23-s + 1.07·26-s − 6.92i·28-s − 3.19·29-s + ⋯ |
| L(s) = 1 | + 0.517i·2-s + 0.732·4-s − 1.78i·7-s + 0.896i·8-s − 1.72·11-s − 0.406i·13-s + 0.925·14-s + 0.267·16-s − 0.662i·17-s − 1.02·19-s − 0.894i·22-s + 0.722i·23-s + 0.210·26-s − 1.30i·28-s − 0.593·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8942509852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8942509852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.732iT - 2T^{2} \) |
| 7 | \( 1 + 4.73iT - 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 2.73iT - 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2.73iT - 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 + 0.196iT - 43T^{2} \) |
| 47 | \( 1 + 8.73iT - 47T^{2} \) |
| 53 | \( 1 + 6.73iT - 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 7.66iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 2.19iT - 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 9.66iT - 97T^{2} \) |
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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
−8.543233581969059020539836542313, −7.894802086818582785703910521430, −7.22196303928244806450182775604, −6.89036674465689126180958792859, −5.65639207195702036337256333143, −5.07052352779606431056583633597, −3.93731722232933425983024296761, −2.96914950105314895806336775733, −1.86519193110810062509929848644, −0.27245872938044513129069274538,
1.87828592089834310639286359870, 2.44722476970710319474752121084, 3.19988725232541530320876935937, 4.54391002278571785271266190062, 5.57301587193185918398847097280, 6.08267121493940639388107077075, 6.99462969290502792791024329986, 8.043174458754633509202254383090, 8.560511118028070499179126669414, 9.457936015258583126760889644229
