| L(s) = 1 | + 1.53·3-s − 0.507·5-s + 2.03·7-s − 0.639·9-s + 3.46·11-s − 2.02·13-s − 0.779·15-s + 2.23·17-s − 1.14·19-s + 3.12·21-s − 7.76·23-s − 4.74·25-s − 5.59·27-s − 5.16·29-s − 6.89·31-s + 5.32·33-s − 1.03·35-s − 5.52·37-s − 3.11·39-s − 9.13·41-s + 3.83·43-s + 0.324·45-s − 8.61·47-s − 2.85·49-s + 3.44·51-s + 10.8·53-s − 1.75·55-s + ⋯ |
| L(s) = 1 | + 0.886·3-s − 0.226·5-s + 0.769·7-s − 0.213·9-s + 1.04·11-s − 0.562·13-s − 0.201·15-s + 0.543·17-s − 0.262·19-s + 0.682·21-s − 1.61·23-s − 0.948·25-s − 1.07·27-s − 0.959·29-s − 1.23·31-s + 0.926·33-s − 0.174·35-s − 0.907·37-s − 0.499·39-s − 1.42·41-s + 0.584·43-s + 0.0483·45-s − 1.25·47-s − 0.408·49-s + 0.481·51-s + 1.48·53-s − 0.236·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 + 0.507T + 5T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 7.76T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 - 8.34T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.66501437086239647266661066622, −7.39939057535794959642101013643, −6.26720736285331959294011772212, −5.60266888101059449179109481217, −4.73540306066737621420601656329, −3.76457478872216956000565308396, −3.47180661040053001470446572293, −2.11128512894345920496906880599, −1.72076781088671770268452009931, 0,
1.72076781088671770268452009931, 2.11128512894345920496906880599, 3.47180661040053001470446572293, 3.76457478872216956000565308396, 4.73540306066737621420601656329, 5.60266888101059449179109481217, 6.26720736285331959294011772212, 7.39939057535794959642101013643, 7.66501437086239647266661066622
