| L(s) = 1 | − 1.72e3·2-s − 5.90e4·3-s + 8.88e5·4-s + 1.02e8·6-s + 3.15e8·7-s + 2.08e9·8-s + 3.48e9·9-s − 4.66e10·11-s − 5.24e10·12-s − 7.73e11·13-s − 5.45e11·14-s − 5.47e12·16-s − 8.80e12·17-s − 6.02e12·18-s − 2.41e13·19-s − 1.86e13·21-s + 8.05e13·22-s − 1.52e13·23-s − 1.23e14·24-s + 1.33e15·26-s − 2.05e14·27-s + 2.80e14·28-s − 2.37e15·29-s + 5.25e15·31-s + 5.07e15·32-s + 2.75e15·33-s + 1.52e16·34-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.577·3-s + 0.423·4-s + 0.688·6-s + 0.422·7-s + 0.687·8-s + 0.333·9-s − 0.541·11-s − 0.244·12-s − 1.55·13-s − 0.504·14-s − 1.24·16-s − 1.05·17-s − 0.397·18-s − 0.903·19-s − 0.243·21-s + 0.646·22-s − 0.0765·23-s − 0.396·24-s + 1.85·26-s − 0.192·27-s + 0.179·28-s − 1.04·29-s + 1.15·31-s + 0.797·32-s + 0.312·33-s + 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.01931388750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01931388750\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.72e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 3.15e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 4.66e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.73e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 8.80e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.41e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.52e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.37e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.25e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.94e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.01e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.67e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.47e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.88e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.82e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.59e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.79e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 8.97e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.61e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.46e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.51e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.37e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.85e20T + 5.27e41T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47219904761527485696448480258, −9.646097595343710525026184051115, −8.548343617795623814444953032688, −7.59502684581041921232580412009, −6.67617892154855236205333417964, −5.12931663625873717500675803956, −4.36883665402390430123010869981, −2.43247922872224017183018687988, −1.54008510676769879924577646616, −0.07178111326604605366664136880,
0.07178111326604605366664136880, 1.54008510676769879924577646616, 2.43247922872224017183018687988, 4.36883665402390430123010869981, 5.12931663625873717500675803956, 6.67617892154855236205333417964, 7.59502684581041921232580412009, 8.548343617795623814444953032688, 9.646097595343710525026184051115, 10.47219904761527485696448480258
