| L(s) = 1 | − 26.9·2-s + 20.4·3-s + 214.·4-s + 1.22e3·5-s − 549.·6-s − 4.89e3·7-s + 8.02e3·8-s − 1.92e4·9-s − 3.29e4·10-s + 9.03e4·11-s + 4.37e3·12-s + 1.14e5·13-s + 1.31e5·14-s + 2.49e4·15-s − 3.25e5·16-s + 5.19e5·18-s + 8.64e5·19-s + 2.61e5·20-s − 9.98e4·21-s − 2.43e6·22-s + 1.60e6·23-s + 1.63e5·24-s − 4.62e5·25-s − 3.07e6·26-s − 7.94e5·27-s − 1.04e6·28-s + 1.73e6·29-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.145·3-s + 0.418·4-s + 0.873·5-s − 0.173·6-s − 0.770·7-s + 0.692·8-s − 0.978·9-s − 1.04·10-s + 1.86·11-s + 0.0608·12-s + 1.10·13-s + 0.917·14-s + 0.127·15-s − 1.24·16-s + 1.16·18-s + 1.52·19-s + 0.365·20-s − 0.112·21-s − 2.21·22-s + 1.19·23-s + 0.100·24-s − 0.236·25-s − 1.32·26-s − 0.287·27-s − 0.322·28-s + 0.456·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.676482598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676482598\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 26.9T + 512T^{2} \) |
| 3 | \( 1 - 20.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.22e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.89e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.03e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.14e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.64e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.60e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.62e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.60e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.74e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.87e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.13e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.53e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.93e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.17e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.46e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.98e7T + 7.60e17T^{2} \) |
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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
−9.841465625794382056475986766124, −9.086189329318118585013399155916, −8.911483698764058144697914277136, −7.48489848458417665600711013461, −6.44301028131673231982428310200, −5.65791716367588776337700401485, −3.99600342116912582588106853002, −2.84179743047647674653424032543, −1.40856828759385088436498149711, −0.78476066572236959785604536903,
0.78476066572236959785604536903, 1.40856828759385088436498149711, 2.84179743047647674653424032543, 3.99600342116912582588106853002, 5.65791716367588776337700401485, 6.44301028131673231982428310200, 7.48489848458417665600711013461, 8.911483698764058144697914277136, 9.086189329318118585013399155916, 9.841465625794382056475986766124
