| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5·13-s + 14-s + 16-s + 3·17-s + 2·19-s − 9·23-s − 5·25-s + 5·26-s + 28-s − 3·29-s + 5·31-s + 32-s + 3·34-s + 2·37-s + 2·38-s − 6·41-s − 43-s − 9·46-s − 6·47-s + 49-s − 5·50-s + 5·52-s + 3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 1.87·23-s − 25-s + 0.980·26-s + 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s − 1.32·46-s − 0.875·47-s + 1/7·49-s − 0.707·50-s + 0.693·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172706901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172706901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−11.61812839447450755099610871682, −10.58318399981159313328624140160, −9.704000550255874231272189290913, −8.360658565008429029479731162281, −7.66081859530557802153309309893, −6.29670817383847541122613475379, −5.61879984028395623126782300752, −4.30645127647783484429164317499, −3.34773629764867655679280376645, −1.68869994047015348511995782895,
1.68869994047015348511995782895, 3.34773629764867655679280376645, 4.30645127647783484429164317499, 5.61879984028395623126782300752, 6.29670817383847541122613475379, 7.66081859530557802153309309893, 8.360658565008429029479731162281, 9.704000550255874231272189290913, 10.58318399981159313328624140160, 11.61812839447450755099610871682
