| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s + 3·11-s + 4·13-s + 4·14-s − 4·16-s − 2·17-s − 2·19-s + 6·22-s + 5·23-s + 8·26-s + 4·28-s + 29-s + 2·31-s − 8·32-s − 4·34-s + 5·37-s − 4·38-s + 41-s + 43-s + 6·44-s + 10·46-s + 6·47-s − 3·49-s + 8·52-s + 3·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s + 0.904·11-s + 1.10·13-s + 1.06·14-s − 16-s − 0.485·17-s − 0.458·19-s + 1.27·22-s + 1.04·23-s + 1.56·26-s + 0.755·28-s + 0.185·29-s + 0.359·31-s − 1.41·32-s − 0.685·34-s + 0.821·37-s − 0.648·38-s + 0.156·41-s + 0.152·43-s + 0.904·44-s + 1.47·46-s + 0.875·47-s − 3/7·49-s + 1.10·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.306476599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.306476599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.918311564552459896414129871883, −6.99219953777008933414075068338, −6.37793889190286755907620802267, −5.84982517389395981659595838466, −4.99342273522673413054819938011, −4.38244342721143014380994810645, −3.83671545156978843468561546559, −3.00570317852034526630587963516, −2.06295820671583417203795756531, −1.01836719130596143283894004150,
1.01836719130596143283894004150, 2.06295820671583417203795756531, 3.00570317852034526630587963516, 3.83671545156978843468561546559, 4.38244342721143014380994810645, 4.99342273522673413054819938011, 5.84982517389395981659595838466, 6.37793889190286755907620802267, 6.99219953777008933414075068338, 7.918311564552459896414129871883
