L(s) = 1 | − 16·2-s − 233.·3-s + 256·4-s + 356.·5-s + 3.72e3·6-s − 4.09e3·8-s + 3.46e4·9-s − 5.70e3·10-s + 7.28e4·11-s − 5.96e4·12-s − 3.93e4·13-s − 8.31e4·15-s + 6.55e4·16-s + 5.11e5·17-s − 5.54e5·18-s − 9.00e5·19-s + 9.13e4·20-s − 1.16e6·22-s + 3.82e5·23-s + 9.54e5·24-s − 1.82e6·25-s + 6.29e5·26-s − 3.48e6·27-s − 4.73e6·29-s + 1.33e6·30-s + 7.93e5·31-s − 1.04e6·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.66·3-s + 0.5·4-s + 0.255·5-s + 1.17·6-s − 0.353·8-s + 1.75·9-s − 0.180·10-s + 1.50·11-s − 0.830·12-s − 0.382·13-s − 0.424·15-s + 0.250·16-s + 1.48·17-s − 1.24·18-s − 1.58·19-s + 0.127·20-s − 1.06·22-s + 0.285·23-s + 0.587·24-s − 0.934·25-s + 0.270·26-s − 1.26·27-s − 1.24·29-s + 0.299·30-s + 0.154·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.7595614054\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7595614054\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 233.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 356.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 7.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.00e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.82e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.93e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.72e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.00e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.69e8T + 7.60e17T^{2} \) |
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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
−11.95621801494012932014941899150, −11.00529087954350185761522192926, −10.12644909036089991879773064990, −9.087745511484869929114577609298, −7.41822321796819488523426894992, −6.36981301071607788447844058256, −5.56960918888171157739784238834, −4.02174750087763800153197190446, −1.72334655430287513571037709543, −0.61081025555080911694194492422,
0.61081025555080911694194492422, 1.72334655430287513571037709543, 4.02174750087763800153197190446, 5.56960918888171157739784238834, 6.36981301071607788447844058256, 7.41822321796819488523426894992, 9.087745511484869929114577609298, 10.12644909036089991879773064990, 11.00529087954350185761522192926, 11.95621801494012932014941899150