| L(s) = 1 | + 0.453·3-s + 2.43·5-s + 1.57·7-s − 2.79·9-s − 1.43·11-s − 3.06·13-s + 1.10·15-s − 17-s + 3.99·19-s + 0.715·21-s − 6.82·23-s + 0.939·25-s − 2.62·27-s − 3.52·29-s + 7.83·31-s − 0.650·33-s + 3.84·35-s − 6.03·37-s − 1.38·39-s + 1.39·41-s + 7.98·43-s − 6.81·45-s − 8.67·47-s − 4.51·49-s − 0.453·51-s + 1.22·53-s − 3.49·55-s + ⋯ |
| L(s) = 1 | + 0.261·3-s + 1.08·5-s + 0.596·7-s − 0.931·9-s − 0.432·11-s − 0.849·13-s + 0.285·15-s − 0.242·17-s + 0.917·19-s + 0.156·21-s − 1.42·23-s + 0.187·25-s − 0.505·27-s − 0.655·29-s + 1.40·31-s − 0.113·33-s + 0.649·35-s − 0.992·37-s − 0.222·39-s + 0.217·41-s + 1.21·43-s − 1.01·45-s − 1.26·47-s − 0.644·49-s − 0.0634·51-s + 0.168·53-s − 0.471·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.453T + 3T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.52T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 6.03T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 + 8.67T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 5.05T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 2.49T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.71646981954710134509323116674, −6.72775155859650486811084108718, −5.97337891393813811270883175097, −5.42365336699682071463907155933, −4.87651905990885825607187883604, −3.88303351750914221930778503460, −2.82477931265470061761084355316, −2.31108928444899979402357407075, −1.48772203313641968810649845750, 0,
1.48772203313641968810649845750, 2.31108928444899979402357407075, 2.82477931265470061761084355316, 3.88303351750914221930778503460, 4.87651905990885825607187883604, 5.42365336699682071463907155933, 5.97337891393813811270883175097, 6.72775155859650486811084108718, 7.71646981954710134509323116674
