L(s) = 1 | + 3-s + 1.28·5-s + 0.120·7-s + 9-s − 5.49·11-s − 3.49·13-s + 1.28·15-s + 4.85·17-s − 1.05·19-s + 0.120·21-s + 7.39·23-s − 3.34·25-s + 27-s + 0.667·29-s − 10.3·31-s − 5.49·33-s + 0.155·35-s − 7.38·37-s − 3.49·39-s + 6.28·41-s + 3.21·43-s + 1.28·45-s + 11.6·47-s − 6.98·49-s + 4.85·51-s − 12.5·53-s − 7.06·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.575·5-s + 0.0456·7-s + 0.333·9-s − 1.65·11-s − 0.967·13-s + 0.332·15-s + 1.17·17-s − 0.241·19-s + 0.0263·21-s + 1.54·23-s − 0.669·25-s + 0.192·27-s + 0.123·29-s − 1.85·31-s − 0.956·33-s + 0.0262·35-s − 1.21·37-s − 0.558·39-s + 0.981·41-s + 0.489·43-s + 0.191·45-s + 1.70·47-s − 0.997·49-s + 0.679·51-s − 1.72·53-s − 0.952·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 0.120T + 7T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 - 7.39T + 23T^{2} \) |
| 29 | \( 1 - 0.667T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 7.89T + 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 0.550T + 73T^{2} \) |
| 79 | \( 1 - 0.308T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 0.397T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
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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
−7.55665417086161869712330006884, −7.07101061318604069882722261572, −5.94720429766388346373194329134, −5.30483072742290235408326910647, −4.89175996450627832111223130784, −3.76021825629360566389156867101, −2.91279365404630935159018986483, −2.39169029208986507564480009199, −1.44515229452879826290113240883, 0,
1.44515229452879826290113240883, 2.39169029208986507564480009199, 2.91279365404630935159018986483, 3.76021825629360566389156867101, 4.89175996450627832111223130784, 5.30483072742290235408326910647, 5.94720429766388346373194329134, 7.07101061318604069882722261572, 7.55665417086161869712330006884