| L(s) = 1 | − 1.75·2-s − 81·3-s − 508.·4-s + 625·5-s + 141.·6-s + 2.35e3·7-s + 1.78e3·8-s + 6.56e3·9-s − 1.09e3·10-s − 5.80e4·11-s + 4.12e4·12-s − 3.56e4·13-s − 4.12e3·14-s − 5.06e4·15-s + 2.57e5·16-s + 5.67e5·17-s − 1.14e4·18-s − 1.30e5·19-s − 3.18e5·20-s − 1.90e5·21-s + 1.01e5·22-s − 1.17e6·23-s − 1.44e5·24-s + 3.90e5·25-s + 6.24e4·26-s − 5.31e5·27-s − 1.19e6·28-s + ⋯ |
| L(s) = 1 | − 0.0774·2-s − 0.577·3-s − 0.994·4-s + 0.447·5-s + 0.0446·6-s + 0.370·7-s + 0.154·8-s + 0.333·9-s − 0.0346·10-s − 1.19·11-s + 0.573·12-s − 0.345·13-s − 0.0287·14-s − 0.258·15-s + 0.982·16-s + 1.64·17-s − 0.0258·18-s − 0.229·19-s − 0.444·20-s − 0.214·21-s + 0.0925·22-s − 0.876·23-s − 0.0891·24-s + 0.200·25-s + 0.0267·26-s − 0.192·27-s − 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.006987804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006987804\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 1.75T + 512T^{2} \) |
| 7 | \( 1 - 2.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.80e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.56e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.67e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 1.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.92e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.50e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.97e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.69e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.56e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.17e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.25e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.64e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.49e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.58e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 5.12e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.02e9T + 7.60e17T^{2} \) |
| 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
−10.07562433337516815632495116233, −9.597965498595450030858799832089, −8.198008341287422152649406619178, −7.62975528640918878870892162563, −6.02187895079481946793027209525, −5.28739911493237841148809055025, −4.49751822542694385919178560191, −3.12820374553830946050281983283, −1.64193662530629540187482100417, −0.48186099471475849484586717465,
0.48186099471475849484586717465, 1.64193662530629540187482100417, 3.12820374553830946050281983283, 4.49751822542694385919178560191, 5.28739911493237841148809055025, 6.02187895079481946793027209525, 7.62975528640918878870892162563, 8.198008341287422152649406619178, 9.597965498595450030858799832089, 10.07562433337516815632495116233
