| L(s) = 1 | − 3-s + 2.21·5-s − 1.52·7-s + 9-s − 11-s + 13-s − 2.21·15-s − 1.73·17-s − 4.28·19-s + 1.52·21-s − 0.474·23-s − 0.0967·25-s − 27-s + 5.11·29-s + 4.21·31-s + 33-s − 3.37·35-s + 0.428·37-s − 39-s + 6.28·41-s + 8.16·43-s + 2.21·45-s − 5.05·47-s − 4.67·49-s + 1.73·51-s − 14.0·53-s − 2.21·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.990·5-s − 0.576·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.571·15-s − 0.421·17-s − 0.982·19-s + 0.332·21-s − 0.0989·23-s − 0.0193·25-s − 0.192·27-s + 0.950·29-s + 0.756·31-s + 0.174·33-s − 0.570·35-s + 0.0704·37-s − 0.160·39-s + 0.980·41-s + 1.24·43-s + 0.330·45-s − 0.736·47-s − 0.667·49-s + 0.243·51-s − 1.92·53-s − 0.298·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 0.474T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 - 0.428T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 8.16T + 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 - 6.02T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 + 0.260T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 0.428T + 97T^{2} \) |
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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
−7.57582558002897098810882007316, −6.49576690627500548943456013452, −6.34624629703255212895480814491, −5.66670241323523675938689987830, −4.78449249919672334061431380048, −4.16186908479613113399229036356, −3.02063622225986590289262711491, −2.26834208228290779198913750183, −1.28745037676339275962905045268, 0,
1.28745037676339275962905045268, 2.26834208228290779198913750183, 3.02063622225986590289262711491, 4.16186908479613113399229036356, 4.78449249919672334061431380048, 5.66670241323523675938689987830, 6.34624629703255212895480814491, 6.49576690627500548943456013452, 7.57582558002897098810882007316
