| L(s) = 1 | + 2.56·2-s − 161.·3-s − 249.·4-s − 415.·6-s − 1.29e3·8-s + 1.96e4·9-s + 4.04e4·12-s − 4.31e4·13-s + 6.05e4·16-s + 5.04e4·18-s + 2.79e5·23-s + 2.09e5·24-s + 3.90e5·25-s − 1.10e5·26-s − 2.12e6·27-s + 5.32e5·29-s − 1.25e6·31-s + 4.86e5·32-s − 4.90e6·36-s + 6.98e6·39-s + 5.13e6·41-s + 7.17e5·46-s − 3.11e6·47-s − 9.80e6·48-s + 5.76e6·49-s + 1.00e6·50-s + 1.07e7·52-s + ⋯ |
| L(s) = 1 | + 0.160·2-s − 1.99·3-s − 0.974·4-s − 0.320·6-s − 0.316·8-s + 2.99·9-s + 1.94·12-s − 1.50·13-s + 0.923·16-s + 0.480·18-s + 23-s + 0.632·24-s + 25-s − 0.241·26-s − 3.99·27-s + 0.752·29-s − 1.35·31-s + 0.464·32-s − 2.92·36-s + 3.01·39-s + 1.81·41-s + 0.160·46-s − 0.638·47-s − 1.84·48-s + 49-s + 0.160·50-s + 1.47·52-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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5563359245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5563359245\) |
| \(L(5)\) |
|
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 | \( 1 - 2.56T + 256T^{2} \) |
| 3 | \( 1 + 161.T + 6.56e3T^{2} \) |
| 5 | \( 1 - 3.90e5T^{2} \) |
| 7 | \( 1 - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.69e10T^{2} \) |
| 29 | \( 1 - 5.32e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.25e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.51e12T^{2} \) |
| 41 | \( 1 - 5.13e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.11e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.52e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 1.91e14T^{2} \) |
| 67 | \( 1 - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.32e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 4.99e7T + 8.06e14T^{2} \) |
| 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
−16.42442292415185327321591804764, −14.80935380423050568115951117362, −12.94340974939646264484421352028, −12.20573812798288630229092135669, −10.78407804078334704513648951630, −9.569195258661682746768575480396, −7.12821196931798717727353152062, −5.48365710965517680196720795192, −4.54563824817196368509283972522, −0.66124939014099614407429649788,
0.66124939014099614407429649788, 4.54563824817196368509283972522, 5.48365710965517680196720795192, 7.12821196931798717727353152062, 9.569195258661682746768575480396, 10.78407804078334704513648951630, 12.20573812798288630229092135669, 12.94340974939646264484421352028, 14.80935380423050568115951117362, 16.42442292415185327321591804764
