| L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·11-s − 13-s + 17-s + 4·19-s + 2·21-s + 8·23-s − 5·25-s − 27-s + 2·29-s + 6·31-s − 2·33-s + 8·37-s + 39-s + 4·41-s + 4·43-s − 3·49-s − 51-s − 14·53-s − 4·57-s − 8·59-s − 10·61-s − 2·63-s + 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.348·33-s + 1.31·37-s + 0.160·39-s + 0.624·41-s + 0.609·43-s − 3/7·49-s − 0.140·51-s − 1.92·53-s − 0.529·57-s − 1.04·59-s − 1.28·61-s − 0.251·63-s + 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590617140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590617140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.67950205935192, −15.83628012328202, −15.69874036144325, −14.88175732306430, −14.22641144445315, −13.68498875595087, −12.99412920314538, −12.50959095501372, −11.96181754874469, −11.28343724918531, −10.91174657813054, −9.955552564402685, −9.557494843827964, −9.152308043756880, −8.107553014669859, −7.534767970266213, −6.819476911770268, −6.260982387527504, −5.705911463754020, −4.837393583908609, −4.286837482071561, −3.281202761502914, −2.770727786647186, −1.471974421445127, −0.6492996418946811,
0.6492996418946811, 1.471974421445127, 2.770727786647186, 3.281202761502914, 4.286837482071561, 4.837393583908609, 5.705911463754020, 6.260982387527504, 6.819476911770268, 7.534767970266213, 8.107553014669859, 9.152308043756880, 9.557494843827964, 9.955552564402685, 10.91174657813054, 11.28343724918531, 11.96181754874469, 12.50959095501372, 12.99412920314538, 13.68498875595087, 14.22641144445315, 14.88175732306430, 15.69874036144325, 15.83628012328202, 16.67950205935192
