| L(s) = 1 | − 1.56·3-s + (−2 − i)5-s + (−2 + 1.73i)7-s − 0.561·9-s − 6.16·11-s − 3.56i·13-s + (3.12 + 1.56i)15-s − 2.70·17-s − 7.12i·19-s + (3.12 − 2.70i)21-s − 7.12·23-s + (3 + 4i)25-s + 5.56·27-s − 2.70i·29-s − 5.40·31-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + (−0.894 − 0.447i)5-s + (−0.755 + 0.654i)7-s − 0.187·9-s − 1.85·11-s − 0.987i·13-s + (0.806 + 0.403i)15-s − 0.655·17-s − 1.63i·19-s + (0.681 − 0.590i)21-s − 1.48·23-s + (0.600 + 0.800i)25-s + 1.07·27-s − 0.502i·29-s − 0.971·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1106435381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1106435381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 3.56iT - 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 2.70iT - 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 + 5.40iT - 41T^{2} \) |
| 43 | \( 1 - 12.3iT - 43T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 2.24iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 4.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 5.40iT - 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.010706772086385431614183603200, −8.252076612393624840192003565698, −7.65386375130012769795103460021, −6.68820087139355751995584838798, −5.74834217159220654526853939902, −5.26351874428576298410617242223, −4.49983604625212083530338710230, −3.18456363627890868209142597341, −2.47999640939055799282120108425, −0.33842098357280075308813145700,
0.12875324407159591626281183767, 2.11133543730756050899126439671, 3.34329107233947475868706082145, 4.04510821697327216669836355543, 5.05159806248120934038998246041, 5.89036935609373018778181013876, 6.61936481990846916450216694792, 7.39135441929906383540382530517, 8.022166305515387330861720808670, 8.879084405783400447473172871025
