| 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
−7.24463743792516860934665498419, −6.83992826132783558430590421709, −6.40148329464394259176704085909, −5.51652713971765912700468215061, −4.89372306449345739942739228477, −4.27611418656492076772928086632, −3.12718992341435283186024061001, −2.38543930015145758462202185692, −0.925668752746411571143809294425, 0,
0.925668752746411571143809294425, 2.38543930015145758462202185692, 3.12718992341435283186024061001, 4.27611418656492076772928086632, 4.89372306449345739942739228477, 5.51652713971765912700468215061, 6.40148329464394259176704085909, 6.83992826132783558430590421709, 7.24463743792516860934665498419
