| L(s) = 1 | − 28.9·2-s + 80.1·3-s + 579.·4-s − 2.31e3·6-s − 9.35e3·8-s − 144.·9-s + 4.64e4·12-s + 5.40e4·13-s + 1.21e5·16-s + 4.18e3·18-s + 2.79e5·23-s − 7.48e5·24-s + 3.90e5·25-s − 1.56e6·26-s − 5.37e5·27-s + 8.68e5·29-s + 1.80e6·31-s − 1.13e6·32-s − 8.39e4·36-s + 4.32e6·39-s − 4.60e6·41-s − 8.08e6·46-s − 6.45e6·47-s + 9.76e6·48-s + 5.76e6·49-s − 1.12e7·50-s + 3.12e7·52-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.988·3-s + 2.26·4-s − 1.78·6-s − 2.28·8-s − 0.0220·9-s + 2.23·12-s + 1.89·13-s + 1.86·16-s + 0.0399·18-s + 23-s − 2.25·24-s + 25-s − 3.41·26-s − 1.01·27-s + 1.22·29-s + 1.95·31-s − 1.07·32-s − 0.0500·36-s + 1.86·39-s − 1.63·41-s − 1.80·46-s − 1.32·47-s + 1.83·48-s + 49-s − 1.80·50-s + 4.28·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.085926960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085926960\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 2.79e5T \) |
| good | 2 | \( 1 + 28.9T + 256T^{2} \) |
| 3 | \( 1 - 80.1T + 6.56e3T^{2} \) |
| 5 | \( 1 - 3.90e5T^{2} \) |
| 7 | \( 1 - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.14e8T^{2} \) |
| 13 | \( 1 - 5.40e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.69e10T^{2} \) |
| 29 | \( 1 - 8.68e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.80e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.51e12T^{2} \) |
| 41 | \( 1 + 4.60e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.45e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.52e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 1.91e14T^{2} \) |
| 67 | \( 1 - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.47e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.83e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.83e15T^{2} \) |
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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
−16.19763589350545722064343956206, −15.17225826493685568459326604326, −13.57358699137694629103351250428, −11.48934113174410361454083235145, −10.26071600882990030330086952673, −8.785528370880907108738112345945, −8.312193720324238660650046295629, −6.61290661913085672954290949270, −2.95312795013608813421629405734, −1.15438384073458975501791693746,
1.15438384073458975501791693746, 2.95312795013608813421629405734, 6.61290661913085672954290949270, 8.312193720324238660650046295629, 8.785528370880907108738112345945, 10.26071600882990030330086952673, 11.48934113174410361454083235145, 13.57358699137694629103351250428, 15.17225826493685568459326604326, 16.19763589350545722064343956206
