| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s + 197.·7-s − 64·8-s + 81·9-s − 100·10-s + 380.·11-s + 144·12-s − 672.·13-s − 788.·14-s + 225·15-s + 256·16-s − 583.·17-s − 324·18-s + 361·19-s + 400·20-s + 1.77e3·21-s − 1.52e3·22-s + 4.67e3·23-s − 576·24-s + 625·25-s + 2.68e3·26-s + 729·27-s + 3.15e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.947·11-s + 0.288·12-s − 1.10·13-s − 1.07·14-s + 0.258·15-s + 0.250·16-s − 0.489·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.877·21-s − 0.669·22-s + 1.84·23-s − 0.204·24-s + 0.200·25-s + 0.780·26-s + 0.192·27-s + 0.759·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.925112979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925112979\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 197.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 380.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 672.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 583.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 47.9T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.57e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.99e3T + 8.58e9T^{2} \) |
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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
−9.703717521133321551427431732676, −9.082379799055656557894335413219, −8.305702445407872471164496798869, −7.43615936007630658961934755231, −6.67179033628088889541865553261, −5.24603368369475360784769810786, −4.39649707150869806329010654328, −2.84020804765044859236916856727, −1.85016339222958260576121931579, −0.975798987156110298495077573061,
0.975798987156110298495077573061, 1.85016339222958260576121931579, 2.84020804765044859236916856727, 4.39649707150869806329010654328, 5.24603368369475360784769810786, 6.67179033628088889541865553261, 7.43615936007630658961934755231, 8.305702445407872471164496798869, 9.082379799055656557894335413219, 9.703717521133321551427431732676
