| L(s) = 1 | + 0.266·3-s − i·5-s − 2.75·7-s − 2.92·9-s + (2.14 + 2.52i)11-s + 4.13·13-s − 0.266i·15-s − 2.18i·17-s + 0.708i·19-s − 0.734·21-s + 0.631i·23-s − 25-s − 1.58·27-s − 3.76·29-s − 2.39i·31-s + ⋯ |
| L(s) = 1 | + 0.153·3-s − 0.447i·5-s − 1.04·7-s − 0.976·9-s + (0.647 + 0.762i)11-s + 1.14·13-s − 0.0688i·15-s − 0.529i·17-s + 0.162i·19-s − 0.160·21-s + 0.131i·23-s − 0.200·25-s − 0.304·27-s − 0.698·29-s − 0.429i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3392662134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3392662134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (-2.14 - 2.52i)T \) |
| good | 3 | \( 1 - 0.266T + 3T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 2.18iT - 17T^{2} \) |
| 19 | \( 1 - 0.708iT - 19T^{2} \) |
| 23 | \( 1 - 0.631iT - 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 + 6.68iT - 37T^{2} \) |
| 41 | \( 1 - 3.61iT - 41T^{2} \) |
| 43 | \( 1 - 6.51iT - 43T^{2} \) |
| 47 | \( 1 + 1.36iT - 47T^{2} \) |
| 53 | \( 1 + 5.57iT - 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 4.31iT - 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 2.69iT - 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
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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
−9.122242978111716441959366563624, −8.184039546451124246777886486701, −7.44357639209325338533852640019, −6.47388059875755683743843341572, −6.01747820685873030971828067308, −5.15990222357015157684696268872, −4.09358016302801447918879613168, −3.44717895967773512480732513201, −2.51518425576426645814098816130, −1.30134844592483418600501046827,
0.10064244051207676305822743660, 1.56625037615407816545996879824, 2.98067661819764082801042010925, 3.33094798066723574174603250630, 4.17256886300596996122356409960, 5.52503350074258477182562388798, 6.18999093947930487555828632526, 6.51226668912143119179271422552, 7.55377842621548726317854485807, 8.452063858945005885013355713355
