| L(s) = 1 | − 3-s − 3·5-s + 10·7-s − 26·9-s − 11·11-s − 16·13-s + 3·15-s + 42·17-s − 116·19-s − 10·21-s − 189·23-s − 116·25-s + 53·27-s − 120·29-s + 163·31-s + 11·33-s − 30·35-s − 409·37-s + 16·39-s + 468·41-s − 110·43-s + 78·45-s − 144·47-s − 243·49-s − 42·51-s + 90·53-s + 33·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.268·5-s + 0.539·7-s − 0.962·9-s − 0.301·11-s − 0.341·13-s + 0.0516·15-s + 0.599·17-s − 1.40·19-s − 0.103·21-s − 1.71·23-s − 0.927·25-s + 0.377·27-s − 0.768·29-s + 0.944·31-s + 0.0580·33-s − 0.144·35-s − 1.81·37-s + 0.0656·39-s + 1.78·41-s − 0.390·43-s + 0.258·45-s − 0.446·47-s − 0.708·49-s − 0.115·51-s + 0.233·53-s + 0.0809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 189 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 163 T + p^{3} T^{2} \) |
| 37 | \( 1 + 409 T + p^{3} T^{2} \) |
| 41 | \( 1 - 468 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 90 T + p^{3} T^{2} \) |
| 59 | \( 1 - 453 T + p^{3} T^{2} \) |
| 61 | \( 1 - 20 T + p^{3} T^{2} \) |
| 67 | \( 1 - 97 T + p^{3} T^{2} \) |
| 71 | \( 1 - 465 T + p^{3} T^{2} \) |
| 73 | \( 1 - 848 T + p^{3} T^{2} \) |
| 79 | \( 1 - 742 T + p^{3} T^{2} \) |
| 83 | \( 1 + 438 T + p^{3} T^{2} \) |
| 89 | \( 1 + 273 T + p^{3} T^{2} \) |
| 97 | \( 1 - 761 T + p^{3} T^{2} \) |
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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
−11.75138161464091586813014593740, −10.86819667725126269723419370672, −9.839935711203667872891477617365, −8.459471022131887884298509979563, −7.81421701318429530310223653815, −6.30310515843386230087150366007, −5.25898999870401162241406913059, −3.89073708224780319969426521052, −2.20257041689034876636629860905, 0,
2.20257041689034876636629860905, 3.89073708224780319969426521052, 5.25898999870401162241406913059, 6.30310515843386230087150366007, 7.81421701318429530310223653815, 8.459471022131887884298509979563, 9.839935711203667872891477617365, 10.86819667725126269723419370672, 11.75138161464091586813014593740
