| L(s) = 1 | − 9.74·2-s + 22.6·3-s + 62.9·4-s + 3.05·5-s − 221.·6-s − 41.0·7-s − 302.·8-s + 271.·9-s − 29.7·10-s − 55.9·11-s + 1.42e3·12-s − 740.·13-s + 399.·14-s + 69.3·15-s + 928.·16-s − 598.·17-s − 2.65e3·18-s − 347.·19-s + 192.·20-s − 931.·21-s + 545.·22-s + 1.84e3·23-s − 6.85e3·24-s − 3.11e3·25-s + 7.21e3·26-s + 658.·27-s − 2.58e3·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.45·3-s + 1.96·4-s + 0.0546·5-s − 2.50·6-s − 0.316·7-s − 1.66·8-s + 1.11·9-s − 0.0941·10-s − 0.139·11-s + 2.86·12-s − 1.21·13-s + 0.545·14-s + 0.0795·15-s + 0.906·16-s − 0.501·17-s − 1.92·18-s − 0.220·19-s + 0.107·20-s − 0.460·21-s + 0.240·22-s + 0.727·23-s − 2.42·24-s − 0.997·25-s + 2.09·26-s + 0.173·27-s − 0.623·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.169196372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169196372\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - 9.66e5T \) |
| good | 2 | \( 1 + 9.74T + 32T^{2} \) |
| 3 | \( 1 - 22.6T + 243T^{2} \) |
| 5 | \( 1 - 3.05T + 3.12e3T^{2} \) |
| 7 | \( 1 + 41.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 55.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 740.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 598.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 347.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.18e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.17e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 866.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.73e5T + 8.58e9T^{2} \) |
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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
−9.169240579100975577011359162525, −8.548253698298887959775565229352, −7.87868952756422056305933907493, −7.21081544872485868965857528088, −6.42439939844161212921976562920, −4.81641222745739056728692459164, −3.41534602446079545986808654113, −2.47640490293755949849519446081, −1.90300006732369059752743954284, −0.54575963923264932176180212985,
0.54575963923264932176180212985, 1.90300006732369059752743954284, 2.47640490293755949849519446081, 3.41534602446079545986808654113, 4.81641222745739056728692459164, 6.42439939844161212921976562920, 7.21081544872485868965857528088, 7.87868952756422056305933907493, 8.548253698298887959775565229352, 9.169240579100975577011359162525
