L(s) = 1 | + 3-s − 7-s + 9-s + 3·11-s − 2·17-s − 8·19-s − 21-s + 23-s + 27-s − 9·29-s − 6·31-s + 3·33-s − 3·37-s − 8·41-s − 7·43-s + 8·47-s + 49-s − 2·51-s − 2·53-s − 8·57-s + 12·61-s − 63-s − 67-s + 69-s + 3·71-s − 2·73-s − 3·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.485·17-s − 1.83·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s − 1.67·29-s − 1.07·31-s + 0.522·33-s − 0.493·37-s − 1.24·41-s − 1.06·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 1.05·57-s + 1.53·61-s − 0.125·63-s − 0.122·67-s + 0.120·69-s + 0.356·71-s − 0.234·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.197353966072232727160204430573, −7.17916942884544115896989137181, −6.72506684684953914324514722477, −5.94347839079244566016051513575, −4.97605053896591892966431252333, −3.96039209883255910266098516474, −3.59702039257181537355963964944, −2.35571516152391442111067955592, −1.64192067616906587565336345160, 0,
1.64192067616906587565336345160, 2.35571516152391442111067955592, 3.59702039257181537355963964944, 3.96039209883255910266098516474, 4.97605053896591892966431252333, 5.94347839079244566016051513575, 6.72506684684953914324514722477, 7.17916942884544115896989137181, 8.197353966072232727160204430573