| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s − 13-s − 2·17-s − 4·19-s − 2·21-s − 27-s − 4·29-s − 8·31-s − 2·33-s + 6·37-s + 39-s − 6·41-s − 4·43-s − 8·47-s − 3·49-s + 2·51-s + 2·53-s + 4·57-s + 10·59-s + 14·61-s + 2·63-s + 16·67-s + 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s − 0.436·21-s − 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.274·53-s + 0.529·57-s + 1.30·59-s + 1.79·61-s + 0.251·63-s + 1.95·67-s + 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.67842785526080, −14.16856598306006, −13.35464665082472, −12.90541062486026, −12.65627312616152, −11.78157480413660, −11.41939537290286, −11.18049653083814, −10.56383323679645, −9.886580744756137, −9.541548121081863, −8.786185939862291, −8.349427049145795, −7.842747743877838, −7.023750810646062, −6.774306986859608, −6.147451646417930, −5.347382176492965, −5.137993333596130, −4.290074515853950, −3.943330517507005, −3.182442161474965, −2.045527845577764, −1.895868123229750, −0.8780069720061518, 0,
0.8780069720061518, 1.895868123229750, 2.045527845577764, 3.182442161474965, 3.943330517507005, 4.290074515853950, 5.137993333596130, 5.347382176492965, 6.147451646417930, 6.774306986859608, 7.023750810646062, 7.842747743877838, 8.349427049145795, 8.786185939862291, 9.541548121081863, 9.886580744756137, 10.56383323679645, 11.18049653083814, 11.41939537290286, 11.78157480413660, 12.65627312616152, 12.90541062486026, 13.35464665082472, 14.16856598306006, 14.67842785526080
