| L(s) = 1 | + 128·2-s + 284.·3-s + 1.63e4·4-s + 6.10e4·5-s + 3.64e4·6-s + 2.09e6·8-s − 1.42e7·9-s + 7.81e6·10-s + 5.52e7·11-s + 4.66e6·12-s + 3.60e8·13-s + 1.73e7·15-s + 2.68e8·16-s + 2.41e9·17-s − 1.82e9·18-s − 3.65e9·19-s + 1.00e9·20-s + 7.06e9·22-s − 1.80e10·23-s + 5.96e8·24-s − 2.67e10·25-s + 4.61e10·26-s − 8.13e9·27-s − 6.60e10·29-s + 2.22e9·30-s + 1.91e11·31-s + 3.43e10·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0750·3-s + 0.5·4-s + 0.349·5-s + 0.0530·6-s + 0.353·8-s − 0.994·9-s + 0.247·10-s + 0.854·11-s + 0.0375·12-s + 1.59·13-s + 0.0262·15-s + 0.250·16-s + 1.42·17-s − 0.703·18-s − 0.937·19-s + 0.174·20-s + 0.604·22-s − 1.10·23-s + 0.0265·24-s − 0.877·25-s + 1.12·26-s − 0.149·27-s − 0.710·29-s + 0.0185·30-s + 1.25·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(4.493015491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.493015491\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 284.T + 1.43e7T^{2} \) |
| 5 | \( 1 - 6.10e4T + 3.05e10T^{2} \) |
| 11 | \( 1 - 5.52e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.60e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.41e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 3.65e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.80e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 6.60e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.91e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.59e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.96e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.42e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.23e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 6.59e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 8.74e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.38e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 2.35e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.05e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 5.95e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 3.68e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 1.21e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 7.08e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 6.90e14T + 6.33e29T^{2} \) |
| 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
−11.27962594408180278150814000989, −10.09788753242917454514471632031, −8.821280987358038325461443074476, −7.81858520343548320284845089659, −6.08281869208083392824617912318, −5.92218063420673484290335219288, −4.19703834673852946641231486884, −3.33139105878499644648933481978, −2.04495357627837077393842847383, −0.881538225606781191100971430398,
0.881538225606781191100971430398, 2.04495357627837077393842847383, 3.33139105878499644648933481978, 4.19703834673852946641231486884, 5.92218063420673484290335219288, 6.08281869208083392824617912318, 7.81858520343548320284845089659, 8.821280987358038325461443074476, 10.09788753242917454514471632031, 11.27962594408180278150814000989
