| L(s) = 1 | + 16·2-s + 55.3·3-s + 256·4-s + 2.26e3·5-s + 886.·6-s + 5.76e3·7-s + 4.09e3·8-s − 1.66e4·9-s + 3.62e4·10-s − 9.65e3·11-s + 1.41e4·12-s − 6.73e4·13-s + 9.22e4·14-s + 1.25e5·15-s + 6.55e4·16-s + 5.35e5·17-s − 2.65e5·18-s − 1.30e5·19-s + 5.79e5·20-s + 3.19e5·21-s − 1.54e5·22-s + 2.36e5·23-s + 2.26e5·24-s + 3.17e6·25-s − 1.07e6·26-s − 2.01e6·27-s + 1.47e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.394·3-s + 0.5·4-s + 1.61·5-s + 0.279·6-s + 0.907·7-s + 0.353·8-s − 0.844·9-s + 1.14·10-s − 0.198·11-s + 0.197·12-s − 0.653·13-s + 0.641·14-s + 0.639·15-s + 0.250·16-s + 1.55·17-s − 0.596·18-s − 0.229·19-s + 0.809·20-s + 0.358·21-s − 0.140·22-s + 0.176·23-s + 0.139·24-s + 1.62·25-s − 0.462·26-s − 0.728·27-s + 0.453·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.408573448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408573448\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 3 | \( 1 - 55.3T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.26e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.76e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 9.65e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.73e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.35e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 2.36e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.93e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.87e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.10e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.71e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.16e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.12e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.33e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.11e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.90e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.36e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.84e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.23e8T + 7.60e17T^{2} \) |
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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
−14.27203761238214725541467691914, −13.43567990448208815244134921870, −12.07501950228324876934192681212, −10.62981507926939951704488281305, −9.382731960742098120132059111881, −7.82488115513257389497274913201, −5.98506718188984797225884385318, −5.06134790947131822950083339280, −2.87261425884292986877075981537, −1.66034081094288870575595064779,
1.66034081094288870575595064779, 2.87261425884292986877075981537, 5.06134790947131822950083339280, 5.98506718188984797225884385318, 7.82488115513257389497274913201, 9.382731960742098120132059111881, 10.62981507926939951704488281305, 12.07501950228324876934192681212, 13.43567990448208815244134921870, 14.27203761238214725541467691914
