| L(s) = 1 | + 2-s − 3-s + 4-s − 0.279·5-s − 6-s − 7-s + 8-s + 9-s − 0.279·10-s + 2.52·11-s − 12-s + 4.24·13-s − 14-s + 0.279·15-s + 16-s + 3.27·17-s + 18-s − 6.18·19-s − 0.279·20-s + 21-s + 2.52·22-s − 6.09·23-s − 24-s − 4.92·25-s + 4.24·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.125·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.0884·10-s + 0.762·11-s − 0.288·12-s + 1.17·13-s − 0.267·14-s + 0.0722·15-s + 0.250·16-s + 0.793·17-s + 0.235·18-s − 1.41·19-s − 0.0625·20-s + 0.218·21-s + 0.539·22-s − 1.27·23-s − 0.204·24-s − 0.984·25-s + 0.832·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 + 0.279T + 5T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + 6.84T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 0.841T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 - 0.100T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 5.04T + 83T^{2} \) |
| 89 | \( 1 - 8.22T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.35115993901055175912697357731, −6.52671857733669246981110431463, −5.96033614043075405263266827094, −5.68518040709277757021007458870, −4.53411298885357492552245533807, −3.84006824533391209300927715088, −3.53536760845159021087378263564, −2.16092473474189723491500388695, −1.41142785419859063433364356718, 0,
1.41142785419859063433364356718, 2.16092473474189723491500388695, 3.53536760845159021087378263564, 3.84006824533391209300927715088, 4.53411298885357492552245533807, 5.68518040709277757021007458870, 5.96033614043075405263266827094, 6.52671857733669246981110431463, 7.35115993901055175912697357731
