| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−10.09854126146590211624669174065, −9.749205121396688870647577836420, −8.280524075884956092273610432842, −7.55727707153378484774141976097, −6.65963768785974737826480192145, −5.36683559095823513937522588289, −4.95639818252911756604562729940, −3.42683606015106566673715003849, −2.07933314479152275364533906758, 0,
2.07933314479152275364533906758, 3.42683606015106566673715003849, 4.95639818252911756604562729940, 5.36683559095823513937522588289, 6.65963768785974737826480192145, 7.55727707153378484774141976097, 8.280524075884956092273610432842, 9.749205121396688870647577836420, 10.09854126146590211624669174065
