| L(s) = 1 | − 3-s + 9-s − 4·11-s + 6·13-s + 6·17-s − 4·19-s − 27-s + 2·29-s + 8·31-s + 4·33-s − 2·37-s − 6·39-s − 6·41-s − 12·43-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s + 4·57-s + 12·59-s − 14·61-s − 4·67-s − 8·71-s + 6·73-s + 8·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526045624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526045624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.256816469561260775976260152951, −7.66992953206976662106474492455, −6.64268206592688123241801422864, −6.12272660064858180701717007987, −5.38330447809102456056328933809, −4.72724833271437520044860936866, −3.71328443772281499151413162803, −2.99496014822873948694993569874, −1.76568985900583467441729691582, −0.71903881838799907808387254131,
0.71903881838799907808387254131, 1.76568985900583467441729691582, 2.99496014822873948694993569874, 3.71328443772281499151413162803, 4.72724833271437520044860936866, 5.38330447809102456056328933809, 6.12272660064858180701717007987, 6.64268206592688123241801422864, 7.66992953206976662106474492455, 8.256816469561260775976260152951
