| L(s) = 1 | − 1.96e4·3-s + 4.10e6·5-s − 1.45e8·7-s + 3.87e8·9-s + 1.26e10·11-s − 1.34e10·13-s − 8.08e10·15-s + 4.18e11·17-s − 2.26e12·19-s + 2.86e12·21-s − 6.52e12·23-s − 2.18e12·25-s − 7.62e12·27-s − 5.65e13·29-s − 2.27e14·31-s − 2.49e14·33-s − 5.99e14·35-s + 1.18e14·37-s + 2.64e14·39-s − 2.34e15·41-s − 5.53e15·43-s + 1.59e15·45-s + 8.83e15·47-s + 9.84e15·49-s − 8.24e15·51-s + 1.05e16·53-s + 5.20e16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.940·5-s − 1.36·7-s + 0.333·9-s + 1.61·11-s − 0.351·13-s − 0.543·15-s + 0.856·17-s − 1.60·19-s + 0.788·21-s − 0.755·23-s − 0.114·25-s − 0.192·27-s − 0.723·29-s − 1.54·31-s − 0.934·33-s − 1.28·35-s + 0.149·37-s + 0.202·39-s − 1.11·41-s − 1.68·43-s + 0.313·45-s + 1.15·47-s + 0.863·49-s − 0.494·51-s + 0.437·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.96e4T \) |
| good | 5 | \( 1 - 4.10e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.45e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.26e10T + 6.11e19T^{2} \) |
| 13 | \( 1 + 1.34e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.18e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 2.26e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 6.52e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 5.65e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.27e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.18e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 2.34e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 5.53e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 8.83e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.05e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 8.00e15T + 4.42e33T^{2} \) |
| 61 | \( 1 - 2.92e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.49e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.42e15T + 1.49e35T^{2} \) |
| 73 | \( 1 + 5.87e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.16e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.60e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 6.52e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 4.29e18T + 5.60e37T^{2} \) |
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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
−14.71406682492937665737149899955, −13.22122205128259449751343173275, −12.05005560206766916607813844130, −10.21733260780450055484933309952, −9.230045786980291995350824721937, −6.73704186758574996945985309417, −5.83801947788400664030771587612, −3.76863277734620255093821430583, −1.77437464943398714352343813840, 0,
1.77437464943398714352343813840, 3.76863277734620255093821430583, 5.83801947788400664030771587612, 6.73704186758574996945985309417, 9.230045786980291995350824721937, 10.21733260780450055484933309952, 12.05005560206766916607813844130, 13.22122205128259449751343173275, 14.71406682492937665737149899955
