| L(s) = 1 | − 3-s + 5-s − 0.732·7-s + 9-s − 1.26·11-s − 2.73·13-s − 15-s − 19-s + 0.732·21-s + 3.46·23-s + 25-s − 27-s − 2.19·29-s + 4.92·31-s + 1.26·33-s − 0.732·35-s + 4.19·37-s + 2.73·39-s + 4.73·41-s + 6.19·43-s + 45-s − 3.46·47-s − 6.46·49-s − 2.53·53-s − 1.26·55-s + 57-s − 9.46·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.276·7-s + 0.333·9-s − 0.382·11-s − 0.757·13-s − 0.258·15-s − 0.229·19-s + 0.159·21-s + 0.722·23-s + 0.200·25-s − 0.192·27-s − 0.407·29-s + 0.885·31-s + 0.220·33-s − 0.123·35-s + 0.689·37-s + 0.437·39-s + 0.739·41-s + 0.944·43-s + 0.149·45-s − 0.505·47-s − 0.923·49-s − 0.348·53-s − 0.170·55-s + 0.132·57-s − 1.23·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 + 0.928T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 4.19T + 97T^{2} \) |
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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
−7.71100449868886700597569809564, −7.33881018523260123913871389899, −6.24908694924913266206824850002, −5.98373620210564573161353804353, −4.89209556028868913419168897654, −4.51342957722350334999320583339, −3.22188625947209696764156220730, −2.45703976040872182094071098321, −1.31177973225248355716976267477, 0,
1.31177973225248355716976267477, 2.45703976040872182094071098321, 3.22188625947209696764156220730, 4.51342957722350334999320583339, 4.89209556028868913419168897654, 5.98373620210564573161353804353, 6.24908694924913266206824850002, 7.33881018523260123913871389899, 7.71100449868886700597569809564
