| L(s) = 1 | − 3·3-s + 7-s + 6·9-s + 2·11-s + 3·17-s + 8·19-s − 3·21-s − 4·23-s − 9·27-s − 5·29-s + 3·31-s − 6·33-s − 10·37-s − 41-s − 5·43-s + 6·47-s + 49-s − 9·51-s + 9·53-s − 24·57-s + 10·59-s + 13·61-s + 6·63-s − 2·67-s + 12·69-s − 9·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s + 0.603·11-s + 0.727·17-s + 1.83·19-s − 0.654·21-s − 0.834·23-s − 1.73·27-s − 0.928·29-s + 0.538·31-s − 1.04·33-s − 1.64·37-s − 0.156·41-s − 0.762·43-s + 0.875·47-s + 1/7·49-s − 1.26·51-s + 1.23·53-s − 3.17·57-s + 1.30·59-s + 1.66·61-s + 0.755·63-s − 0.244·67-s + 1.44·69-s − 1.06·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−13.62838786785686, −13.49478359949326, −12.63373037120370, −12.12045109391217, −11.95826169983203, −11.39470190584508, −11.26079180897393, −10.31490177678584, −10.13768830703755, −9.739142573632529, −8.974539677743964, −8.454000299328504, −7.667045613890767, −7.268039074205496, −6.849225652324157, −6.251966027722459, −5.585780316300678, −5.364028693429806, −4.999801718348603, −4.030963043731067, −3.849149765444453, −2.987633714183999, −2.000089290036724, −1.315739448158455, −0.8796875608937944, 0,
0.8796875608937944, 1.315739448158455, 2.000089290036724, 2.987633714183999, 3.849149765444453, 4.030963043731067, 4.999801718348603, 5.364028693429806, 5.585780316300678, 6.251966027722459, 6.849225652324157, 7.268039074205496, 7.667045613890767, 8.454000299328504, 8.974539677743964, 9.739142573632529, 10.13768830703755, 10.31490177678584, 11.26079180897393, 11.39470190584508, 11.95826169983203, 12.12045109391217, 12.63373037120370, 13.49478359949326, 13.62838786785686
