| L(s) = 1 | + 3-s − 2·7-s + 9-s + 2·11-s + 2·13-s + 6·17-s + 8·19-s − 2·21-s − 4·23-s + 27-s + 8·29-s + 2·33-s − 10·37-s + 2·39-s + 2·41-s − 12·43-s − 3·49-s + 6·51-s + 10·53-s + 8·57-s − 6·59-s + 2·61-s − 2·63-s − 8·67-s − 4·69-s − 4·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.436·21-s − 0.834·23-s + 0.192·27-s + 1.48·29-s + 0.348·33-s − 1.64·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s − 3/7·49-s + 0.840·51-s + 1.37·53-s + 1.05·57-s − 0.781·59-s + 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s − 0.474·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.830138573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830138573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.30955522254228694838792930706, −9.855081450656987646239307434163, −8.956494428187784870875332301622, −8.065360238159720695767955399926, −7.15109339700237674664040328399, −6.20989492713295015084463586012, −5.14891659899478257407149456127, −3.68478607534762393615004752252, −3.07956734277475342949311345221, −1.32041558029711473446320898964,
1.32041558029711473446320898964, 3.07956734277475342949311345221, 3.68478607534762393615004752252, 5.14891659899478257407149456127, 6.20989492713295015084463586012, 7.15109339700237674664040328399, 8.065360238159720695767955399926, 8.956494428187784870875332301622, 9.855081450656987646239307434163, 10.30955522254228694838792930706
