| L(s) = 1 | + 3-s − 7-s − 2·9-s + 3·11-s − 13-s − 7·17-s − 21-s + 6·23-s − 5·27-s − 5·29-s − 2·31-s + 3·33-s − 2·37-s − 39-s + 2·41-s − 4·43-s − 3·47-s + 49-s − 7·51-s − 6·53-s − 10·59-s − 8·61-s + 2·63-s + 2·67-s + 6·69-s + 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s − 0.218·21-s + 1.25·23-s − 0.962·27-s − 0.928·29-s − 0.359·31-s + 0.522·33-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.437·47-s + 1/7·49-s − 0.980·51-s − 0.824·53-s − 1.30·59-s − 1.02·61-s + 0.251·63-s + 0.244·67-s + 0.722·69-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.610113716136751744673815020975, −7.71169390110110859831183707512, −6.84868577839913740225832418072, −6.30449045983008656645459540885, −5.29174886563521887540154085702, −4.37606541842962934803595143148, −3.48744089310847421866892408097, −2.69661210339499443876888627676, −1.68813850036432900847535927863, 0,
1.68813850036432900847535927863, 2.69661210339499443876888627676, 3.48744089310847421866892408097, 4.37606541842962934803595143148, 5.29174886563521887540154085702, 6.30449045983008656645459540885, 6.84868577839913740225832418072, 7.71169390110110859831183707512, 8.610113716136751744673815020975
