| L(s) = 1 | − 2.23·3-s + 2.82·7-s + 2.00·9-s + 2.23·11-s + 6.32·13-s − 5·17-s + 2.23·19-s − 6.32·21-s − 5.65·23-s + 2.23·27-s − 6.32·29-s − 5.00·33-s − 6.32·37-s − 14.1·39-s − 3·41-s − 8.94·43-s − 2.82·47-s + 1.00·49-s + 11.1·51-s − 12.6·53-s − 5.00·57-s + 8.94·59-s + 6.32·61-s + 5.65·63-s + 11.1·67-s + 12.6·69-s − 14.1·71-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 1.06·7-s + 0.666·9-s + 0.674·11-s + 1.75·13-s − 1.21·17-s + 0.512·19-s − 1.38·21-s − 1.17·23-s + 0.430·27-s − 1.17·29-s − 0.870·33-s − 1.03·37-s − 2.26·39-s − 0.468·41-s − 1.36·43-s − 0.412·47-s + 0.142·49-s + 1.56·51-s − 1.73·53-s − 0.662·57-s + 1.16·59-s + 0.809·61-s + 0.712·63-s + 1.36·67-s + 1.52·69-s − 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.63546108685782837301332365840, −6.64992967509686784162895911886, −6.31226789177240133421034519567, −5.53224794620438483930413436677, −4.94536079570409028357950511873, −4.14003135956332946957860444703, −3.46229444715692074087110569494, −1.88927430765790092051176333887, −1.31214932400486549167291312364, 0,
1.31214932400486549167291312364, 1.88927430765790092051176333887, 3.46229444715692074087110569494, 4.14003135956332946957860444703, 4.94536079570409028357950511873, 5.53224794620438483930413436677, 6.31226789177240133421034519567, 6.64992967509686784162895911886, 7.63546108685782837301332365840
