| L(s) = 1 | + (−1.35 − 0.408i)2-s + (1.66 + 1.10i)4-s − 3.40·7-s + (−1.80 − 2.17i)8-s + 2.19i·11-s − 6.77·13-s + (4.60 + 1.39i)14-s + (1.54 + 3.68i)16-s + 6.90·17-s − 1.49·19-s + (0.895 − 2.96i)22-s + 5.21i·23-s + (9.16 + 2.76i)26-s + (−5.66 − 3.76i)28-s + 2.08·29-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.289i)2-s + (0.832 + 0.553i)4-s − 1.28·7-s + (−0.637 − 0.770i)8-s + 0.660i·11-s − 1.87·13-s + (1.23 + 0.371i)14-s + (0.387 + 0.921i)16-s + 1.67·17-s − 0.343·19-s + (0.190 − 0.632i)22-s + 1.08i·23-s + (1.79 + 0.542i)26-s + (−1.07 − 0.711i)28-s + 0.387·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5642302255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5642302255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.408i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 2.19iT - 11T^{2} \) |
| 13 | \( 1 + 6.77T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 - 5.21iT - 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 - 5.06T + 37T^{2} \) |
| 41 | \( 1 + 1.10iT - 41T^{2} \) |
| 43 | \( 1 + 3.60iT - 43T^{2} \) |
| 47 | \( 1 + 11.0iT - 47T^{2} \) |
| 53 | \( 1 + 1.22iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 - 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 - 1.40iT - 89T^{2} \) |
| 97 | \( 1 + 8.19iT - 97T^{2} \) |
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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
−9.521087653764157221173371081059, −8.335643714287706516516355933411, −7.44384914115574755895989287809, −7.08294451040495010138463742027, −6.09699954511262608764369310736, −5.11236881667520047940164606166, −3.74476381380048331679895652818, −2.94151203755821012541996663987, −1.98572366735055424290108939084, −0.37115213808636371590332934039,
0.878896934525477909413873368099, 2.57583062952009645804165959861, 3.13341557032144284695752731463, 4.66444307414271488462381900669, 5.75589596283039881161232126454, 6.34047822327465349359844515000, 7.23434662565895759868666319737, 7.80829300549169295022785263920, 8.739242788801039204472042363411, 9.543500567190962583280494321454
