| L(s) = 1 | + 512·2-s + 3.80e4·3-s + 2.62e5·4-s + 8.70e6·5-s + 1.94e7·6-s + 1.18e8·7-s + 1.34e8·8-s + 2.82e8·9-s + 4.45e9·10-s + 6.78e7·11-s + 9.96e9·12-s + 5.14e10·13-s + 6.04e10·14-s + 3.30e11·15-s + 6.87e10·16-s − 3.47e11·17-s + 1.44e11·18-s − 1.21e12·19-s + 2.28e12·20-s + 4.48e12·21-s + 3.47e10·22-s − 2.40e12·23-s + 5.10e12·24-s + 5.66e13·25-s + 2.63e13·26-s − 3.34e13·27-s + 3.09e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.11·3-s + 0.5·4-s + 1.99·5-s + 0.788·6-s + 1.10·7-s + 0.353·8-s + 0.242·9-s + 1.40·10-s + 0.00867·11-s + 0.557·12-s + 1.34·13-s + 0.782·14-s + 2.22·15-s + 0.250·16-s − 0.710·17-s + 0.171·18-s − 0.862·19-s + 0.996·20-s + 1.23·21-s + 0.00613·22-s − 0.278·23-s + 0.394·24-s + 2.97·25-s + 0.950·26-s − 0.844·27-s + 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(9.700101613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.700101613\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 29 | \( 1 - 1.45e13T \) |
| good | 3 | \( 1 - 3.80e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 8.70e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.18e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 6.78e7T + 6.11e19T^{2} \) |
| 13 | \( 1 - 5.14e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 3.47e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.21e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 2.40e12T + 7.46e25T^{2} \) |
| 31 | \( 1 + 2.03e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.03e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 9.12e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.97e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.29e16T + 5.88e31T^{2} \) |
| 53 | \( 1 - 4.15e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 1.05e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 9.16e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.45e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 3.51e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 7.59e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 7.84e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 9.00e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.87e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.36e18T + 5.60e37T^{2} \) |
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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
−11.32313800924138495624030933895, −10.28449775533442409503926892345, −9.011352897225014126524950190995, −8.262759775474502941004127580930, −6.53891642268005089752342700944, −5.65597843057612789146153867761, −4.45189155327867191497489636617, −2.98749671688620145214301259191, −2.00257088153994036909552991234, −1.48187590442965528404226886115,
1.48187590442965528404226886115, 2.00257088153994036909552991234, 2.98749671688620145214301259191, 4.45189155327867191497489636617, 5.65597843057612789146153867761, 6.53891642268005089752342700944, 8.262759775474502941004127580930, 9.011352897225014126524950190995, 10.28449775533442409503926892345, 11.32313800924138495624030933895
