| L(s) = 1 | − 2.30·2-s + 3.31·4-s + 2.09·5-s + 3.60·7-s − 3.03·8-s − 4.83·10-s − 11-s + 2.44·13-s − 8.32·14-s + 0.366·16-s − 3.44·17-s − 3.95·19-s + 6.96·20-s + 2.30·22-s − 0.849·23-s − 0.595·25-s − 5.64·26-s + 11.9·28-s − 6.57·29-s + 3.69·31-s + 5.22·32-s + 7.93·34-s + 7.57·35-s − 7.31·37-s + 9.12·38-s − 6.37·40-s + 1.36·41-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.65·4-s + 0.938·5-s + 1.36·7-s − 1.07·8-s − 1.53·10-s − 0.301·11-s + 0.678·13-s − 2.22·14-s + 0.0916·16-s − 0.834·17-s − 0.907·19-s + 1.55·20-s + 0.491·22-s − 0.177·23-s − 0.119·25-s − 1.10·26-s + 2.26·28-s − 1.22·29-s + 0.663·31-s + 0.923·32-s + 1.36·34-s + 1.28·35-s − 1.20·37-s + 1.47·38-s − 1.00·40-s + 0.213·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 3.95T + 19T^{2} \) |
| 23 | \( 1 + 0.849T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 1.36T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 9.31T + 59T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 + 1.57T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−7.891434315577904886222123327579, −7.32128370137473709117337001924, −6.43572606603239694616321543301, −5.84874626253176686417059603420, −4.91653717425026087723445050104, −4.11503870130177707358915114578, −2.66614426337431770222215976962, −1.81329266254869614951347237511, −1.48755758827704879203405734711, 0,
1.48755758827704879203405734711, 1.81329266254869614951347237511, 2.66614426337431770222215976962, 4.11503870130177707358915114578, 4.91653717425026087723445050104, 5.84874626253176686417059603420, 6.43572606603239694616321543301, 7.32128370137473709117337001924, 7.891434315577904886222123327579
