L(s) = 1 | − 3-s + 7-s + 9-s + 3.77·11-s + 13-s − 19-s − 21-s − 23-s − 27-s − 5.77·29-s − 2.77·31-s − 3.77·33-s − 10·37-s − 39-s − 1.77·41-s − 6.54·43-s − 13.5·47-s − 6·49-s + 4·53-s + 57-s − 5.54·59-s + 14.3·61-s + 63-s + 6.77·67-s + 69-s + 4·71-s − 8.22·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.13·11-s + 0.277·13-s − 0.229·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.07·29-s − 0.497·31-s − 0.656·33-s − 1.64·37-s − 0.160·39-s − 0.276·41-s − 0.997·43-s − 1.97·47-s − 0.857·49-s + 0.549·53-s + 0.132·57-s − 0.721·59-s + 1.83·61-s + 0.125·63-s + 0.827·67-s + 0.120·69-s + 0.474·71-s − 0.963·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 8.22T + 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 9.54T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−7.53186218036612498292634474896, −6.73291314670666836012495961430, −6.36258803531372095752363065582, −5.39678666756207341768331403822, −4.90386151464720743676856069677, −3.90532695067768534477287650204, −3.43397834935069854675058224160, −1.99203232109652274594001411782, −1.36368080275610826229873452629, 0,
1.36368080275610826229873452629, 1.99203232109652274594001411782, 3.43397834935069854675058224160, 3.90532695067768534477287650204, 4.90386151464720743676856069677, 5.39678666756207341768331403822, 6.36258803531372095752363065582, 6.73291314670666836012495961430, 7.53186218036612498292634474896